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CHARTS AND GRAPHS 


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CHARTS AND GRAPHS 


AN INTRODUCTION TO GRAPHIC METHODS 
IN THE 
CONTROL AND ANALYSIS OF STATISTICS 


By 


KARL G. KARSTEN, B. A. (Oxon.) 


CONSULTING STATISTICIAN 


INTRODUCTION BY CARL SNYDER 


CHIEF STATISTICIAN OF THE FEDERAL RESERVE BANK OF NEW YORK 


“Tt may be laid down as almost a fundamental principle 
that the statistician who is to be successful in business must 
cultivate the graphic methods.”’—Leonard Ayers. 


p 
, } 
PRENTICE-HALL 
iE ire) 


New York 
PRENTICE-HALL, INC. 
1925 


Copyrighted, 1923, by 
PRENTICE-HALL, INC. 
All rights reserved. 

First printing, October, 1923. 
Second printing, January, 1925. 


PRINTED IN THE UNITED STATES OF AMERICA 


To 
E. D..K.. and E..C, K. 


In inadequate acknowledgment. 


8 


PREFACE 


In its general structure, this book follows a philosophic, 
and not an encyclopedic, arrangement. It therefore inci- 
dentally supports the author’s theory of a system of natural 
evolution of charts, in accordance with which all chart-forms 
fall into line with simple origins and clear channels of growth. 
In the light of this theory, there is no baffling heterogeneity, no 
confusion of purposes or principles, in all the immense multi- 
tude of existing graphic forms. On the contrary, that multi- 
tude resolves itself into a consistent, organic body of simple 
root-forms and logical combinations and developments. Not 
only can we allocate each form to its proper place in such a 
system, but we can often discover gaps in the system, and bring 
to light forms which, while not yet in use, have reason to be. 
As examples of such experience, the following more or less 
original methods may be mentioned. 

The author is indebted to a host of friends and associates, 
and therefore cannot, in a wide’ sense, claim originality for 
any of the methods described. But, in so far as no plain trail 
leads from some of them to any individuals, and, in so far as 
their first use is believed to be made either in this book or in 
earlier work of the author, he may take some small responsi- 
bility for the summary-chart, the double-probabilities paper, 
the population-map, and the collapsible bead-map. The theory 
of the silhouette-bar curves, and the argument for the reversal 
of axes of ogives fall into the same class, as does the use of 
square-root paper for economic data, and the use of the names 
“‘amount-of-change” and “rate-of-change.” ‘The same state- 
ment holds true regarding the compounded-average-seasonal 
method. Needless to say, these are, almost all, inevitable 
results of the application of the theory that chart-forms are 
naturally and logically evolved, one from another. 

The greatest contribution to chart-making, from any single 
source, is the Gantt Progress Chart. This chart is, unquestion- 
ably, the most powerful graphic device for business and for all 


vil 


vill PREFACE 


executive and managerial purposes. While the description has 
been rather full, as given herein, it is by no means complete; 
and the Gantt charting methods, in all their co-ordinated 
ramifications, constitute an independent system of accounting 
and of executive control, which goes far beyond the proper 
field of this book. The present volume must, therefore, be 
supplemented by another, Mr. Wallace Clark’s “The Gantt 
Charts,” to get the full benefits of the method. Mr. Clark’s 
achievements in industrial engineering and the promotion of 
managerial efficiency are ample recommendation for his book. 
And the chart which has been so unqualifiedly praised and 
adopted by Mr. Fred J. Miller, former President of the Ameri- 
can Society of Mechanical Engineers, by Mr. Walter N. 
Polakov, a leading authority on power engineering, and by 
European experts, needs no endorsement from statisticians. 

Inadequate mention has been made in the text, of the work 
of Professor William F. Ogburn (‘Social Change’) on the 
geometric trend of human culture and civilization, which has 
gone far to influence the present writer in his presentation of the 
law of organic growthas the great raison-d etre of rate-of-change 
curves. Professor Ogburn’s careful and keen pioneer work 
in this field will have an increasing effect upon economic 
thought for a long time in the future. 

In a different field, the work of Mr. Carl Snyder should be 
referred to in any discussion of chart methods as it has set a 
high standard in statistical research, and has, by graphic inter- 
pretation, given to abstruse economics a vital and practical 
bearing upon business and commercial welfare. He has been 
a leader in bringing mathematical skill, economic research, and 
business problems together. The student of chart-making can 
do no better than to study the methods used in the charts 
appearing in the Monthly Review of the New York Federal 
Reserve Bank, from which, as will be seen, we have drawn 
heavily for illustrative material. 

Fewer, but as excellent, are the charts appearing in the 
Bulletin of the Cleveland Trust Company, prepared under 
the direction of Dr. Leonard P. Ayers. These charts, and 
Mr. Snyder’s, are models. The charts of the Harvard Bureau 
of Economics, though of a single type, are always powerful 
and well-made. The charts in the Monthly Survey of Current 
Business, published by the Department of Commerce, are 
also well-drawn. Indeed, the use of good charts is steadily 


PREFACE | ix 


increasing. We have seen few books so well illustrated with 
excellent charts as Dr. Ayers’ “The War with Germany,” or 
Mr. Joseph E. Pogue’s “Economics of Petroleum.” 

Others to whom acknowledgment should be made, not alone 
for contributions to this particular volume, but for important 
contributions to the growth of a sound and efficient charting- 
practice, are Mr. John Wenzel and Mr. Arthur R. Burnet, 
both of whom, with the author, earlier enjoyed the privilege of 
working with that pioneer in the field, Mr. Willard C. Brinton. 
To many other economists and former associates, among whom 
may be mentioned Professor Robert E. Hale, Professor Paul 
Douglas, Mr. Stuart Chase, Mr. Paul Brissenden, Dr. Fred R. 
Macauly, Mr. John Scoville, and Mr. Richard Webster, 
the author is indebted in innumerable ways. The courteous 
permission of authors of books and articles in the same field, 
to borrow illustrations from their works, is appreciated, and 
the attempt has been made invariably to credit the sources of 
such illustrations as they appear in the text. It is a pleasure 
and a duty to recommend such important books as those of 
Lipka, Peddle, Running, Haskell, and Brinton; also the 
statistical treatises of Yule, Bowley, Secrist, King, and Kelley. 

A word may be said as to the style of the text. It is a 
quaint and curious folk-way of the academic world that a 
technical account is worthy of respect directly in so far as it 
can not be understood. This hoary tradition is not limited 
to college walls—the rocky road to business, until recently, 
has rested on the self-same supposititious secrecy, and the paths 
of all professions lead to inner circles that guard, as best they 
can, the knowledge and the standards of their work. When 
such precautions make for better craftsmanship, they are most 
heartily to be endorsed. But, when they merely further 
selfish ends, they are a plague and pestilence, and those who 
practice them, only that their own minute monopolies of craft 
may be entrenched, come, sooner or later, into the class of 
parasites, retarding the growth of their profession. 

Having confessed so little patience with the doctrine of 
the incomprehensible per se, we have naturally sought to 
empty the entire bag of tricks, and to tell the whole story 
of the chart in the simplest words that we command. Our 
belief has been that it is a lesser sin to be too easily understood 
than never understood at all. But at the same time, we 
have sought to make the story full and complete. If any of 


x PREFACE 


our readers find charts which do not fall into place in this 
account, but would appear to have been omitted, we beg that 
they will freely advise and assist us to include them. It is, 
moreover, probable that, in spite of vigilant revision, many 
errors have crept in; we hope that readers who detect them will 
courteously co-operate by sending corrections, suggestions, 
and criticisms. 

Chart-making is an art which all can practice. But there 
will always be a world of difference between the charts of 
amateurs and those of master-statisticians. Perhaps the day 
is not far off when, from the latter class, will come a group 
collectively intent on keeping up the standards, not the secrecy, 
of graphs and all statistics. The need for some criterion, high, 
but not too high to be effective, has been already felt, and 
efforts to establish safe statistical standards are on foot. It 
is our understanding that we may shortly look for sets of 
standard texts and examinations, from a committee of the 
American Statistical Association, under the chairmanship of 
Mr. Malcolm C. Rorty. Such steps will be warmly welcomed 
in the profession. The task is to set good standards and to 
make them public property in good plain everyday English. 
And as a contribution to the protection of the calibre of busi- 


ness statistics through the medium of graphic presentation, 
this book is offered. 


Karl G. Karsten 
New York City, September, 1923 


TABLE OF CONTENTS 


xl 


: Page 
Pnecocuctions by ‘Carle snyder. £504. exec coda oka es oe XXXVI 
BOOK I. SIMPLE CHARTS 
Part I. Non-MatTuemaTicaL CHarTs 

Chapter Page 
tw vlaps andabiaerams eunanceR atid. 1 

Tl. Classification-Chartsxi : 3. i 13 

Pye ovte-Chattey on &.avcisedl tae vAl| 
TV. Composite Chaftse*:..0< 39) 

Part IJ. Amount-oF-CHANGE ANALYSIS 

Re POE ALISTICS 5. 134.5055 testhoe ne rath eat Ms Ba ak 48 
VI. Work-sheets.. Sop 
VII. Co-ordinates. . 63 
VIII. Dimensions aad Warn bless ee da the 
De Hundred-PerCent Bats.>... cu... 2 36- 83 

Oy Boe 1e-Ae i aheay ee Petes ch ee ee ial cs ede 89 

XC Bat-Charts. . LES eae a 99 
XII. Composite Bar-Charts.. 111 
XIII. Pictorial Bar-Charts.. 124 
XIV. Vertical Bar-Charts . 134 
DON ML CUPVCSGeey cacao ivnete deeds ee i ete ir d's 
VI. de oom en eee 154 
DOVES CaleS oie hs hee. ae we wee ote L6F 
Poy Tlie sr oreime=Lomtss At ai nah. eee ce... “190 
pat Ga Composite- Curves «cadens nao 6 facta oe ps 198 
Dodane Historical Curvess-eeee ee aeceie occ s. 220 
POxte Cycles a... heptane rd Ape ates 235 
Dae ear 7pe-RALts i wise ee be. ets. -aba.! (252 
Pe eliemee rogtess-© watts. ks Warnes. «2 ie | 2OL 
POs OUI aly ee atts tae Air et oniiks ate «+s 278 


CONTENTS 


Xi 
XXV.’- Silhouette Bar-Chatts.<... ise eas 285 
SOX VED “index: Numbers: te ac oe os eee 294 
MXVIIO “Frequency Seties 00.44. so: is soe ens ee 308 
OV iliserequency Curves: 02> vce ec ee 326 
BX OBI ES ciate oem ies err can tie Ore wharage 341 
Ne LORehe CALVES 6.15 toe ee ait boos 356 

BOOK II. ADVANCED CHARTS 
Part III. Rare-or-CHaNncE ANALYSIS 
Chapter Page 
XXxXI. The Genealogy of Numbers.............. 366 
XXXII The'Law of Organie Growth ©. 03.60.24)... SLY 
XXXIII. Rate-of-Change Analysis........ 382 
EXOCK LV. cRatesof-Change Sealesois i Pee. 3. 5 ee 
XXXV. Rate-of-Change Curves.................. 402 
XXXVI. Historical Rate-of-Change Curves......... 416 
XXXVII. Logarithmic Frequency Curves........... 426 
POCXV IT. « Lomasthnne Ogivesic. ix eee a oe 444 
Part IV. Spectra, ANALYSES 
XXXIX: “The: Normal'Curve of Error 2.09). 9.5 450 
XL. Probability Curves. . D0 454 
XLIL. Shifted Zero-pointsrcht hc. eet cee 472 
XLIT...“Curve:Fiecing) cee eis co ee eee 477 
XLII. Specially Projected Scales. . 482 
XLIV.. Formulae for Gives. cna opee eee eee 490 
Part V. CatcuLatinc CHartTs 

XUV. Curves for Formulae’y: 4.2 ee 511 
XLVI. «Parallel Nomogtaphs». 2.) eee 55 
XLVII. Zigzag and Composite Sit UM ae 560 
XLVIII. Slide-Rules.. S77, 
XLIX. Hundred- PerCent Trianples 588 


CONTENTS xiil 


Part VI. Two- anp Turee-DIMENSION Data 


L. ‘Hundred-Per-Cent Squares............... 598 
PA ireaeiar-Ghiatts: 00s 7. fe cere is avec. 613 
Pl iy ePopilatrons Maps.) iis... oo Ee ea ay 623 
LIII. Models... RM rag hs et meth a 50) 
EV the Third Dimension. . Len Mea eae OOS 
LY: Frequency Surfaces...................... 650 
ey Weep eet aS tie. Mian kes hehe ces OGL 


Part VII. Conc.usion 


LVII. The Statistical Materials. . Me Bor eee eC OKl 
Pyle he, Function of Charts: .<f6) ees 5660. 684 
APPENDICES 

AppENDIx A. Implements for Making Charts......... 691 

AppEenpIx B. Steps in Making Charts. Pa eken a ROOO 

Appenpix C. Methods of een Charts so ns ve 102 

Aprenpix D. Colors in Charts. es hee eee LOD 

Aprenpix E. Optical Illusions in TOhare ost Cael 

MePENDIXAN. . ehe:Verbal Charts. .eescc-0¢ ve cae. 713 
BIBLIOGRAPHY 

BereerTS ER TOGRNPUY 1. ceo ec tla cubase dee eae LL 

INDICES 
INDEX OF PERSONS AND SOURCES. i eer en 1a, 
INDEX OF ILLUSTRATIONS BY SuBJECT Matter... Soak? 


GENERAL INDEX. Ry a et oat one Oar eke age ee Phen, ne Ran 


.. 


77 


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SERCO STON CUS ONS) Faas 


LIST OF ILLUSTRATIONS 


By CHART-METHOD 


PAGE 
BICTORIAL MAP ORSTHE UNITED! STATES iainnci nels taescinee ccc o6e 2 
HEART-SHAPED MAP OR THE WORLD «. .c0ceeecscecsssereacs Pa ae 3 
MER eATOR/SeEROJECTIONSOF THE WORLD © a5. Go00 cs. 2 osc aaleates ea) 4 
HEMISPHERICAL PROJECTION OF THE WORLD..........c0+csccceceeece 4 
PLEIPTICAL PROJECTION OF -THE® WORLD 9. s4d.40c 4 occa ea seine esis vie 5 
HoMoLoGrRaPHIc PROJECTION OF THE WORLD...........e.ceceeeeeees 5 
ID) ATRAMKO ROA FIL OOR=PISA Niue ey chicie sR iis trie, de oes rsh steadin bo asad see 7 
SAMPEESHOR. CROGS-RUDED RAPER ooo violets vio a iocses o avbie aczrsrals oved Aeacebtroupncdre 7 
INNBIONEUNEGHIE Ds LOU OORSBLUAINI ii «ayers casiere hrc ah ches aurore Oa ae Cate Sue 8 
SESE OORSPIANGe MINESH ED eta: Saf ar icnwitiels Cease mo «ioe ooraceeh o's 10 
‘ee NOMENCDATURE* OF) CO-ORDINATES: oc. 0c ieee oe eo oem camecene veel 11 
PNG SIMPIFE RIS OX=GIUART Scars Verio cia dite acct oa orci rs ei um) oes. Cw due ayaa Ro on oe 14 
CHARTAWITH DOXES: OF» VARIOUS. SHAPES o.). 5 aos ood oele «cles Gatenien sis 16 
RADIAEING ORMELANE TAR Vn HART sco bets acie crow vices ap aie.o custome ae 17 
ANSE XAMPTLE | OF COMPLICATED! DATA: 4.5 0hs1del6 a. aS een een os uae 18 
Five INTERLOCKING CLASSIFICATION CHARTS.......<.0..:.2:----.4-- 19 
BIRCH AR Tr Ae eke Rd Se ry (Cea oe AR IO eed Epo Bh ena ten, Sn 20 
A TaBuLaTION_OF StmpLE RouTe-cHarT DatTa..............0.00000- 22 
FAB CONDENSED WiORK=SH EET oy 20s: otis rel ese ach ea nen 23 
AB VERVACONDENSED WORK-SHEET,.) ..c-055.. 012 cisore a oicis, sets avers tian sim aad sions 24 
MiPeOMir EE Ste ROCEDURE<CHART sha nite. wee Gian Sen ae ete ata ee 25 
A StmpLe PRocEDURE-CHART WITH Many ITeEMs...............+-000- 26 
ABRICTORTAP UE ROCEDURE-CHART: 16s55 6 teen nein Se ee ATS nie ee Dif 
AEGRAPHIC OUTLINE TOE WL HOUGHTS hit a de ene ee eta tae 28 
IAS POPULAR, PRESENTATION. ....4..+-.60.- pts” cane wilt ae atl ae 29 
ANSE XCELDENTD Ub ICTORIAL ROUTESCHARD saqsjsscoergecseiscte eeee ee eiee 30 
imiPeANALOGY (OF) VATS, LANKS OR, RESERVOIRS... o0.0000664.4-1 005: 31 
ARSIMBLIETED A GILERETH OROCESS-CHART etapa men eee teed oe ee: 32 
SES IRER CE UOR Nice Fe Se A TN Ue Soyer IEE PO Aes GS oe 33 
AGIM Ee ICECORDs) Ute NOTVAMWLIME-CHAR Tomy. s oe eiee ca chicracoiem cies « 34 
PAW GU AN TORR COLLAR Te rare wet, cies ee Mensa td in oie PRN ees A-ha: wet cca cha en Yenc ea 35 
RESIMPMEMLINE <CHART once tet cen ete tanner Tao Ap vera aiecke aeupecie ean 36 
PVE EE Vall OCKSCIHART a aren Ferenc tee edocs Sloe Shoe eie hace aime whe S17 
PANE UNI G AIH Gl OCR=CHIAR'T: iyo, sar inas co eenee hoke ae wicds secuibcedehs anise een ee 38 
ROUGE SNS Bie eae ce Te ee EO cot a oe ests ne Voncuacaumasnea 40 
TIEN TAD ESP aon Pe CALA SAS Pe EG ordi ie erste ral chslaicutic atenneseisi 41 
FANT SONTEERTCHD RA WAN GHEE ON eet eee eth ne Pitan Neus inion sitesi aya ooue nish 44 
RGUFINGRONT Al OLASSIFICATION-CHART ru iiiimei os a sieiwerie serena <r 45 
(OUEASSTEICA TION-CHAR TVA NDMINVUAD yr ceaie iteie ees cher eore acti co aio) acres ais aie secs 46 
IABSIMPLES COMPUTING LOHEET cor ipiectecn ras ie me Shee sion ereiae aainne sane 54 
Various GEOGRAPHIC GROUPINGS OF THE STATES..........--+--+00+> 56 
An INCOMPLETE STATE-LIST GEOGRAPHICALLY ARRANGED..........+-- Sh 
GUAGSIFIED) HEADINGS TO. THEN COLUMNS asi odin ce eee es 59 
Cotumn Symsots, FoRMULAE, AND CLassirieD Stuss AND CaptTions.. 60 
CoLuMN SYMBOLS AND COMPUTING INSTRUCTIONS. ...-s+eerseereeeees 61 


XV 


LIST OF ILLUSTRATIONS 


PAGE 
Sreps IN LAYOUT OF CO-ORDINATES ...... Pes ere Cee Ota are errs aioe 63 
Ay | eee er en ere ira amen eh ors ae MaeD ODS Soe TO con SO 64 
GR eno 2s cas gree Sohogs a bien eines SRR Loh sear Sets ota aueatepeltan enetent Me et eter of enc omen tt 66 
OA 9 ri eal Connon Ae adeeb La el aang hei SO an eo a aUIG C 67 
MIMADIA AEE IOOLOLUD INONA Ie rigs pan gcnanSmalhon at opavovomnmdonma does 68 
SCAT RS VOR AXES => UNEQUAL; saiteriedonieh aris tetas asl meena tar erereet ietaere 68 
ORIGIN OF CHARTI NEAR ONE EDGE Om EIELD) aya tase eee eit 70 
QORIGIN-IN (CORNER: OF TED cance nia eae erare tei oc ie ae eee eto 70 
ORIGINS NOT. SHOWN IN” CHAR I tin oe aetna eres cain ee ceee iene tees Si Oi 71 
Fretp witH Co-orpinaTES Not PERPENDICULAR........-..--+++eeee> 71 
Tue Turee Axes oF THREE-DIMENSIONAL SYSTEM OF PERPENDICULAR 
COsORDINATE SS Sifart chee octire ees rere oe ae nara a oth eens een UP 
POLAR CO-ORDINATES eae bcp ein ee oe oie ene tees eae hoe eee reine 72 
Two Lines or Bars, OnE Twice as Lonc as THE OTHER..........- 78 
Tue AREA OF THE SECOND SQuARE IS Four TIMEs THE First....... 78 
Tue AREA OF THE SECOND SguaRE Is Twice THAT OF THE FirsT.... 79 
Tue AREA OF THE SECOND Circle 1s Four Times THE FirstT........ 79 
Tue AREA OF THE SECOND CIRCLE Is TWICE THE FirRST............. 80 
Tue HeIcHT oF THE SECOND FicuRE Is TWIcE THE FIrsT..........- 80 
Tue Heicuts oF THE Two Ficures ARE IN THE RaTIO OF ONE TO THE 
SousRERoor or Iwo.ee or: oe ee ee 80 
Tue Errecrs of ComPpaRISON BY LINEAR, SQUARE, OR CuBic Measures 81 
(ASSmmprn 1009" DAR Mao vas n con oct ecco thee ee aioe cen eee ere 83 
MANY SEGMENTS--INO LOM ADING dict eae ker hit merce terete tate ree eS 84 
CLASSTIFICATION-CHART TANDNLOUG, AR a mies amie esietcier cerita aor 85 
DISTINGE! SHADING Goch: fone foto tike chee he hora erae eee ne renee 86 
Won BARSIARE” PASILY: COMPAREIN. cc tra oe crrccetnesicneienen een Serene 86 
CoMPARISON, OF DHREE, DIFEERENT SY EARS soe ssos cies oe eae eee 87 
AT SIMPLER? LO0GG* CIRCEES yew neues cater: erratic io ae cee eee 90 
ACCURATE, COMPARISONS’ CANNOT EE IVIADE ee lencen os een 91 
THE Less) DETraiisaiHe BERTERG cies een cee oe ee eee 92 
LABELUING VS) IDIFFICUD T=“ om oes caer cntociccse oe cree CTR Een 93 
EXCELEENT DOLLAR. CHARTSo) onda ee ere ee ee eee 94 
SHADING THE SEGMENTS TO INCREASE POPULARITY.............+ee0+- 95 
How, 10; CALIBRATE THE CIRCLES. oon eee Bre 82, Se ee ee 96 
MANY SEGMENTS=s INQ LSHADINGs o 0 seen coe aie cere terre eens ae 97 
ASPIE-CHARTY TINS IVLR TAMA f 5 creer cine oak leer ett aE ee 97 
As SIMPLE BARSCHART: pancho bacon wee ee Oe ETO ETT 99 
DETAILEDMD ATA MAYS Bw INCUUDED sree nyt ere caeeeiee eter tena 100 
A LONG BAR BROKEN TOTSAVED SPACE.) (aon cee ee ace ce nner eee 101 
NATIONAL DistRIBUTION BY STATES AND STATE-GROUPS.........----- 102 
AN: ALPHABETICOARRANGEMENED toa): ieee ciccith ners amcnietiertieta rene cient 103 
HISTORICALS DATA. WLUST BEAINE ORDER ar teen Umer eterna 104 
Pracinc tHE Mosrmilmportann EURsT os a2 Geen ae eee eee 105 
THE ARRANGEMENT IN ORDER OF SIZE 1S POPULAR........2.-+e00eee 106 
‘LHe GAnrr IpEENESS-CHART soon rte eee oe een ee 106 
CUASSIFICATION-CHARTO AND IB AR-CHART gra) enen cian ieee eee 107 
THe Tora. BAR ISGWIDER oe eS ee ee eee 108 
An Orrice Recorp Form To Inctupr BARs.......................- 108 
Bars As PART OF THE OFFICE RECORD 4 a) een eee 109 
TYPEWRITTEN BARS EROR. TYPED VESieerre ee nce 110 
‘Tne CompounD BAR-CHARTD ey en ee en een 111 


Tue Cuarr Dors Nor Surrer FROM DetalLep Statistics ATTACHED.... 112 
VERY SMALL, SEGMENTS VIAN Em SEO WiNtetonn aa enero enter nn ne 113 


LEST OF TELUST RATIONS Xvii 
PAGE 
Tue RELaTiIve (OR PERCENTAGE) BAR-CHART.....ccccccsccceccccucee 114 
Any Pair or 100% Bars Reatty Forms a Revative Bar-cnart.... 114 
MEMO UETIPLE BARSCHART woe yu. Ska oon cctlec ds coke. eter OP 115 
A Goop Comparison oF Historica, DaTa........ GS tieeat eh ete 115 
CorRRELATION 1s INDICATED BY MIRRORING........0.-.00ccecceuceuse 116 
DMMMEERY HAS ONEY VAN POPULAR VWALUEM ulcers heds nar coe ee 117 
Connection Lings or SHapincs To DistinGuisH SEGMENTS.......... 118 
WarPING THE CHART TO SHOW THE TREND OF CHANGES............. 119 
Connection Lines. (Nore InserTeD DATA)..........eccecccecsce- 120 
A Compounp Muttipie (ABpsoLuTe) BAR-CHART..........eeceeeeeees 121 
A Compounp MuttipLe (RELATIVE) BAR-CHART.......-..e00ceeeeees 122 
Tue Compounp RELATIVE Is THE BEsT OF THE ComposITE BAR-CHARTS.. 122 
Tue Stmpter Forms Are More EFFEcrTIve.............-.+++--.... 123 
Tue Circtes Must Have Unirorm Rapit....... Arenas Tipe eens peas te 126 
SEGMENTED, LIKE THE COMPOUND BAR-CHART.........0-ecceceeeeces 127 
SUGGESTING UNETA TE COINS arora ta oe, ae face ec we AE ae 127 
PicrortaL Figures May Be SuBsTITUTED FOR BARS..........e+eee0. 128 
sires Dairy SDIMENSTONSIS ORNAMENTAL). vcs.) -suscee Sie sto eicac.e ences 129 
One or Many Devices to Stimurate INTEREST................-.--- 130 
PornTERS, INSTEAD OF SEGMENTS, SUGGEST PRESSURE GUAGE DIalLs..... 131 
AEROPLANES, HorsE-RACES, BOATRACES, AND THE LIKE, HAVE A CERTAIN 
RORMUNRIC NCAT UES eee nn ts a7 ese ee Me Cee eg 22 late 132 
FAMVIR TIGAD MARA CHEA RU ater «28 aps chuck ava op acs elk ony aeances AEs soci lore oatee 2 134 
AE SERENSOED VERTICAL EARS a fc te ae sihsts Ge otaarnd okt Ears Osean 135 
ISARECETAR THWILHWA PR TELD iat ssc ceehrn cbs otvager seats o.oo Sinai bce eae ae 136 
THe VERTICAL BaR-CHART WITH Data Reapinc UpwarpD............ 137 
irre ACARIS VOR D WASILYS ROAD o . ceitiis Aled state css usuaethe Aamo 138 
IABREDATIVEGNLUL TIPE ES INR@CHAR TT cet cede so oheoe vse ca aiaie erate eee ees 139 
Note THE Key To THE SHADING ACCOMPANYING THE DaTA.......... 140 
IAN ABSOLUTE VLUIMIIPL ED DAR-CHART Ao cuis.s Sick e sdesanenine aetna eon aa 141 
Wipe Bars with Data INSERTED........ Ai Peele eit tars oie eit | Seen vas 142 
CONNECTING LINESTARE OFTEN WSEFUD..c.cau-u le sa tds sea. 143 
Connectinc Lines Usep In ComPpaRISON OF Two DirrerenT Years... 144 
FUER E SISRUHES DATA—-TISTORICAIA : c.cic cashes on cvtee aie e hicee Sarasa ce aie 145 
MAES ORDINARY? DARSCHART OR onriaie 3) ide sisrciners os ouis siaeiontic rneceetele 146 
WERTICADM BOARS ROR] ROPULARITYs1.). facies oie ete te vere ee 146 
ARGCGRVED LHROUGHOTHE | DARS 1 ag) rien -nilnccie aacisuseee acto Gattscie 147 
Tue Bars DisAPPEARING; THE “FIELD”. APPEARING.............----- 147 
ime EVOLUTION OFSTHES CURVE IS. COMPLETE. 5 = <csrc siete amrciv eine cere te 148 
LIER E MISE OAT SONO TOLN TE OERLE Sica este citer: ao Prat one nen SCL: 150 
INGLCURVELSHOULDERE IVIADE WITH: DHIS® 25. scuueesgs > ocoaren ses ae 151 
aE AMPUTATED CHART. 15) DECEPTIVE vei nth cic cee nni oe cre es 154 
Mires GAGERA GAINS TS AMPUTATION) 16) CLEAR <aeriienes & Aomawaeeeee 155 
Wuen ZERO 1s ARBITRARY, IT CAN BE OMITTED..........-+00-0000: 156 
Sietee WV HITECCONESVVARNS: LHE, READER) oe seteaan nae ks one 158 
Tue UNEVEN BaseE-LInE I[npvicaTEes Tuat 17 1s Not THE REAL BASE-LINE 159 
A Wavy BaseE-LINE IS A SHORT-HAND WARNING..........+----eeeeee 159 
Time) ZERO-LINE) SHOULD ALWAYS BE) HEAVY. sass + eee. seeset es - ssa. 160 
Tur Sounp PosiTIoN FOR Two VERTICAL SCALES.......-0-++eeeeeee 162 
An INTERESTING CoMPARISON OF DiF¥YERENT PEeRIODS............... 163 
Tue VerticaL AND HortzonTaL SCALES ARE EQUAL.............-... 167 
Tur HorizontaL SCALE 1s Twice THE LENGTH OF THE VERTICAL SCALE 168 
AMCURVETONTAN LIELD LWITH OM OUAL, SCALES es iis csanele cekeuy iene evita 168 


LIST OF ILLUSTRATIONS 

PAGE 

A Curve on A Freip wits THE HorizonTaL ScALe Twice THe LENGTH 
OF-THE VERTICAL SCALE =. cicnae ine cies eet eee a mene ere 169 
Tue VERTICAL SCALE IS TWICE THE LENGTH OF THE HorizonTat ScaLe.. 169 
A CurVE ON THE FIELD WITH THE LonG VERTICAL SCALE.......---- 170 
A Comparison OF CURVES DRAWN ON DiFFERENT VERTICAL ScaLes... 171 
ExampLes OF CoNVENIENT HorizonTAL SCALES. ee et i rt 17/5: 
SHow1nc One Monru sy Days on LETTER-SIZE “Papen. ; ee 174 
One YEAR BY Montus ON LETTER-SIZE PAPER.........--:eseeeeeeee 175 
ONE DECADE- BY” YEARS oct eee A Setar eins Ronee setae ares 176 
One OQUARTER-CENTURY BY YEARS»= oie cic ante tse oe eet aloe Wize 
One YEAR BY WEEKS ON DouBLE LETTER-SIZE PAPER......-.+++000- 178 
One YEAR BY WEEKS ON LETTER-SIZE PAPER......-----eeeeeeeees 179 
SEVERAL CHARTS OVERLAID APPEAR AS ONE... .. 0.00. ee crete cece 180 
Tue Freak Peak NEED Nor BE ACCOMMODATED........-...-++++eee- 181 
Draco BE CHARTED: .c. see ae oe eee Ae Ce eee rar 182 

Tue Hicuesr Pornr IN THE SERIES SHOULD BE PLOTTED aBouT Two- 
THIRDS: UP THE) CHART=FIELD np mae ee Oo eee ee eee eae 183 
COMMERCIAE PoORMS AVAIUABLE. 102.2% <scete oes cee crete 184 
To Ostain A SCALE SMALLER THAN THosE GiveN By THE RULER.... 185 
ENGINEERS” [URIANGULAR RULED Goose cue oe oe eee Soe ie ie areas 186 
To Ostain A SCALE LarGER THAN THOSE GIVEN BY THE RULER........ 186 
EXAMPLES OF CONVENIENT’ VERTICAL SCALES...-..--c..4-5) asses 187 

TABLE FOR VERTICAL SCALES WITH ENGINEERS’ RULES ON 6-, 8-, AND 
LOFINCH. RETDS23 ae Oak ots eile ae ee ek ee ie Rt it eee 188 
A. CHAR T+FLEED ccc os5< & 5:deeio spat als Se ane ee ade eee 190 
To Pror ANYWHERE BETWEEN ORDINATES.......-.....:0:2+ess-+-e- 192 
LotPror Onry Upont ORDINSTES oe eee eee ee ae eee 193 
DATA: WITH DIFFERENT INTERVAES 0 coo ce aoe ee eee eee 194 

INTERPOLATION FOR THE PERIOD OF THE WAR AND EXTRAPOLATION FOR THE 
YEARS) AFTER 1919: Seis ce Sees 196 
EXTRAPOLATION; ooo c.5.0 bls hc oars ee ee ee 197 
EAch Curve “Hassirs OWN VERTICAL SCALED ae oes ee oer 198 
Eacu Curve Has Irs Own Horizonrat SCALE.......---seeeeeeeees 199 
Tue Heavy Ling 1s Usep ror tHe More Important CurVE.......- 200 
ZONES: INSTEAD: OF (CURVES! o6 Go. e ks oe es eee 201 
AN) EXCELLENT: FORM OF ZONE-CURVER In; arate tee er neae 202 
It 1s UsELEss To SHow ALL THE INDIVIDUAL CURVES.............+> 203 
AN EXcELLENT ADAPTATION OF THE ZONE-CURVE .......--.eeeeeeeees 204 
A. Gun-sHor CHart’: oreo oe ee ee eee eee 205 
Tue Staircase Curve 1s NEAR To A BaAR-CHART. F ARR Se 206 

An AxssotutE Compounp PIPE-orGAN eee. OR AN ABSOLUTE 


STAIRCASED, BAND-CHART..0 he nee Sec ne ee 207 
Tue SMooTHED AND StarrcaseE Curves DirrerR IN OuTLINE AND Areas 208 


PSEUDO-sSTAIRCASED CURVES 2 A0e Ga eee eee 209 
SEVERAL PskuUDO-sTAIRGCAGED. CURVEST Opin ee a ce eee 210 
IAGPSEUDO=STAIRCASED) CURVE eric tenn ate omer tne ae 211 
AN INTERESTING Usr or SHADINGS IN A BAND-CHART OR VERTICAL BAR- 
CHART 505 donors ccc y ae arr a 211 
Tue Curves ARE TRUE ONLY FOR CUMULATIONS OF THE LAYERS........- 212 
A Retative (on PercentaGe) BAND-CHART..........ceseecveeteewes 213 
eae SMOOTHED Reiative BAND-GHART 4 ¢d.0ccs4> oe. ce ane eee 214 
‘He STarcaseD RetaTive BAND-CHART.....-..+c1-+.0-0s seuss seuss 215 


AEOMOOTHED RELATIVE. BAND-CHART Rien cnet nic ee ene 216 
A Smoornep Retatrye BaNnp-CHART... ee DOT 


FIG. 


198. 
199, 
200. 
201. 
202. 
203. 


204. 


205. 
206. 
207. 


208. 
209. 
210. 
211. 


DV 
2S, 
214. 
215. 
216. 
Ne 
218. 


219: 
220. 


221 


Wipe 


223. 


224. 
225. 


226. 
221, 


228. 
229. 
230. 
Zon. 
232. 
233. 
234. 
235. 
236. 
237. 
238. 
259; 


240. 
241. 


242. 


243. 


244, 
245. 
246. 
247, 


248. 
249, 
250. 


LIST OF ILLUSTRATIONS XIX 


PAGE 
AN ExceLLent BAND-CHART (ABSOLUTE)... .ccesccceccusacevcucs aoe 218 
Tue RELATIVE CHART Is SUPPLEMENTARY ....0.0.0ccccecucccuceccces 218 
PRI CTORTALOCURVE She phd A wee ae ae VE 219 
PETES TORIEAE. SERING TON aye? Mice os bac ee ateone N D. 220 
PPAR BY Nilontis, ONTVERSAL RULING, (0. 0.c6. ivan, cccdu.ccaeaeh.. 221 
ae INDIVIDUAL CHARTS COMBINE EASILY <4... sos csccsun seems - 222 
Fanninc Up anp Down To Compare SEASONALS..........0ee-cee-. 224 
SmmpLE SERIES AND ANNUAL CUMULATIONS...........0--c0ccecceecee 225 
SERIES AND CuMULATION PLotrep WirH Same SCALE................ 226 
arse DATAG OANNOT BEL CUMULATED. — o60 8. hock be eee. 227 
Tue Srmpce Series anp Its Movinc ANNUAL TOTAL................ 228 
THREE PosITIONS FOR THE SAME MoviNG TOTAL...........-..e000-- 231 
ADELA MOMUTIEMICAGTORIGURES: <-) iiin onc ek te Ea de bleed wake 232 
Movinc Annuat ToTaL AND AVERAGE SERIES...........-.eeeeeeee 233 
Tue Movinc ANNuaL AVERAGE GIVES THE TREND.............----. 234 
spe VR CHANICAL TC YCLOGRAPH a iacns ceiikic cre Cee en es Peean 235 
FASPAG CHART “HAISuISMNVORTHLESS «006 a0.ck cio artic ead cot Uae been 236 
Tue ReEcTILINEAR Co-orDINATES ARE Mucu Betrer................ 237 
A FREE-HAND AND ImaGINARY PIcTURE OF THE BusinEss CYCLE...... 238 
CYCUESTOR SLIGHTLY. VARYING LENGTHS. a4. 6s 00 eo ee oe eee Hee 239 
SHOWING THE UsE oF RELATIVE (PERCENTAGE) FIGURES AND A ROUNDED 
GRE eS cies ects hs nce Eee Ca eae ee eye ene ae 241 
YATE CCVICUE Shela bisnees Shetek ane Ree cek, | COREL Rim i Sei 242 
ComPaRISON OF Two or More Cyciic PEeRiops ON ‘ONE CHaRT...... 243 
V.CEEGEIVUAVERCTIANGE rt tc eae een yal a hace dae Lr eedsw. re is eet wake tokens 244 
Mi RENIOVING AVERAGE SSHOWS “LREND esi. oiess) alps keens te ensie whine © caveleictetelel« 245 
Tue SEASONAL CycLE CoMPUTED FROM THE OnE Cyc tic Periop........ 247 
(DHE CSEASONAL COMPUTED FROM THE- DREND..:....0.7.4.-onacvussss es 248 
A REMARKABLE Case oF CHANGING CycLE FLUCTUATIONS........... 248 
rE COMPOUNDED AVERAGE.) SEASONAL ss) Gi ccie ees ciate ois eieleiete tla t 250 
ARNG SEMAINE ICG err ere rier ea ee at oe lela Ohutnaked coetacs 253 
Four ZEE-CHARTS ForMING A SINGLE SERIES.........c0seceeeeeeeees 254 
Two Cuarts FANNED OuT For SuccessIVE YEARS.«.......0ee0eeeees 256 
Two Cuarts FANNED Upwarp To Stupy SEASONALS.........+---05: DY 
SRABEEMFOR® CEESCHART (SCALES 905% oly sie os eres ies iralaiers ale SaeieleieGietelte = 258 
NRE DURNET SUARRANGEMENT Moe) tr tore ris cts eco oe ners aud conde Se lence s 260 
DETAIL OF A PROGRESS-CHART FORM—BLANK..........eeeeeceeceees 264 
IDARTETOR DATA FORTA ICROGRESS“CHART ch) .2 2 sees cewe neem erie’ 265 
Tue Quota 1s ENTERED For A YEAR AHEAD.......-..0eeeeeeeceees 266 
ME ROHARTAONRVANUARY (SSD re --- alesadars nicyeveqeletn tasiem tiene ceeioreieloreusiehe © 267 
irre OAR TOONS EBRUARY: COTM o » ahetaicsguc csiss ia citl> is sivis. slepavmsieisin el iets 268 
are ONE ON VLAR CHS UST tis. caine cla < ittet cine or Gt anetereri bia Sere 269 
IMMRROGRESS-CHART OF INALES (BY. DISTRICTS os necie eh el civeid si cieislclivienysy« 270 
A PRocREss-CHART Usep ror A NaTIONAL INVENTORY..........-+--+ 272 
RENCIENS HARE WUSED-IN’ THE WORK=SHOP s..cieucc. os ose acts sine + soleil 274 
DaTA on A SHorT FLy-SHEET—IHE GANTT WAY.........---eeeeeeee 275 
EDIE O CIO RNG OODS sie ete on cco dion rele inoue note reveic take elle Wicc tel ste eiete e's 278 
IMAC ICAU MIN DUSTRIAL © EROCESS visnctais sisiercicis cre «folnitaieie rier sin tetelseheteheactene’s 279 
Bisa ES UINENEA RV SCH ARTS Inne) dic ae, orn a vohe orarorare ie oc onl a ouopers Neier aceseasieietehe:s 281 
IMMN ew ORTEM CHA RTRORM Meier Gite Gree cris meas b atcleedondl “ha, alesoustoyenate:s 282 
A CusroMaRyY AND SouND CoMBINATION OF Bars AND CuRVES....... 284 
KEGURVELIS. TOOMDETAILEDPAND LARGE seme areciatescardre a cceleleke 285 
SEMEN S CRON ETAT ME) ACTA SON ante tt tice heen gore Sadiveverene ciated ansie Mekecadend 85) a 
838 


MEE SAN ER OURVE SEEN FROMPLTS SOND caitayiise ties coin Gai apdiatin holt as 2 


LIST’ OF TLLUST RATIONS 


PAGE 
(Ad DETALDEDGE ORM). Salccas cine o oho ace acets erent onelg) ean ott cen ana fon Reese Yetta 289 
DATA INTHE CHART Saeco eiee iene ie: PAD oie mE ERE Ae he oD 290 
SimpLeE SILHOUETTE Bars PRESENTED HORIZONTALLY.......-+..0+-++: 291 
A Stenovetre Bar-cHART SeT HorizoNTALLY........--52--s0sesece- 292 
A Comparison oF Two Curves ON THE SAME CHART........-.----- 296 
Ne COMPARISON OR SEVERAL) CHARTGs 1 o ceicmic int) sole aed etter tye el areas 297 
Orviousty, Onty InpeEx NUMBERS ARE POSSIBLE...........+--++00-- 298 
Various InpICES OF THE SAME PHENOMENON PRopucED BY DIFFERENT 
METHODS OF “WEIGHTING: —.. cn se) eae coe es te Wente dine eee 299 
PV WA See ee REE AP Pe CR Ae Mba een ree ei Stoney cll) 
dh RE ort eh ete ES ka AH et ta mR PMR es 4 oan era Riad ce he) oihodrt, 301 
i eae ORR OOD mee ie Rint Geberi er pales ep A Me he et tah Stud. whois 8 302 
SE ee eee TERE he het: ot See er ee acne acinomae 303 
CORRELATION SHOWN BY. WIIRRORING. 7 caer: Noches cian etree 304 
ASTIGHTLY.-~LAGGED: CORRELATION: 1 seinen nine ie aici ene 305 
SPL REN SORA SW REPRE ERENT APMC TA rep An OER BB NER mah ey Oath Fn 306 
Ratan S See ra ASM RETO Riee EPA Pee Na era TaN A AE a AC ie ce 308 
oe tn ee ie en ME RAR a Pi kat ORO tae Lol eR Bathe tae. tba ti cid oan 309 
RS Ws 5 ie re me tein 2 Hin a bE SERRA lr ore ois orvid CDs Gomme 6 6 OG tI mE 309 
Teel pee ROR rR EIN OR et htt toRtN eh 3A) Sond eed den to ns sai renga iibc-oe 310 
Tue Raw MarTERIAL FOR A FREQUENCY SERIES.........+-eeeseeeees 311 
ANOTHER) CRUDEM EIST Ms is Gy coe cote, ena Aires aioe ole errr oie 312 
Tue First Step 1s ARRANGEMENT By MaGnITUDE.............----- 313 
AN COMMON, FENDENCY “TO BUNCH) Ube set dete: sede eet etn ee 314 
Pirine UPplon THE (ROUNDS NUMBERS s 1.5 <src chtet recreate nae iene 315 
ComPaRISON OF FouRTEEN SERIES DERIVED FROM THE SAME DaTA BY THE 
Use or DirFERENT Group Limits AND Group SIZES............. 316 
ComPARISON OF CurRVES OF THREE SERIES DERIVED FROM THE SAME Data 317 
‘THE. FREQUENCY "SERTES.. cotsen.t dot tte Ce ne eee ere 319 
Pane RE an oe Peace Nama werj asl hse A ge: ee Ft en i fa nie goin 320 
PEON a See Se GN Fie ah Meee it ee gn CNA ec a NR et 321 
BY RE MGEs coho Canale Swank Cand eee eer Ee a es en ee ee 321 
Bada Sse iocabeounen lel & outa eee eta MEE ae eae ee ae 322 
Sighs guisstinee drove Roop hee be SRC ETC PEE Tae Ten eee 323 
Pa a ee nen eer Re ey Rei Rt I ih ee Fs oy oe 324 
PERIOD" DATA. 5 ao de clo oe, Phen ee ee eee 327 
PERIOD? DATA) C0 cine ee es eo ee eee Ree 328 
PERIOD DATAG i. Paecrtae eta tte ee nr ee 328 
POINT-AND-PERTOD) DATA tate hee ee a ee ee ee 329 
POINT-AND-PERIOD: DATA“. caucus ee ee ee eee 329 
CoMPARISON OF A STAIRCASED AND A SMOOTHED FREQUENCY CuRVE... 330 
THE STAIRCASED: Form 1s APPROPRIATE) 0.0) en cee nee 331 
Ay VERY-SIMPLEST: STATIRCAGED OURVE sient tics atts Sie eae nes 331 
Tue SMootHED Form ts NECESSARY..............000..0.-0 005 sik: a8 332 
Ir 1s Dirricutt To Compare Two Staircase CurVES...........+:-- 333 
A‘ CUMULABLE "SERTES Sth toe ree ree ee ee 334 
AS INON=CUMULABLES SERIBS at eto See a en eee 335 
AS RounDED? CURVE. cn ry en ne ee ee 336 
Computep Averacres Must se Usep ror THE IRREGULAR INTERVALS.. 337 
AoZONED FREQUENCY CURVE. . 4.4 ot sade ee en ee 338 
ASFREQUENCY BAND-CHARTS 2 Coser e hele ee eee ene ee 339 
A Retative Freguency Curve....... Werccren rata ee i AS 340 
aise, ““Less-THAN’ CUMULATIONT at eaten eee eae enn 341 


FIG. 
303. 


304. 
305. 
306. 
307. 
~308. 
309. 
310. 
JAN. 
a12. 
313 
314. 
315. 
316. 
B17; 
318. 
S19. 
320. 
321. 
322. 
323. 
324. 


325. 
326. 
BL 
328. 
329. 
330. 
331. 
D2: 
333. 
334. 
335. 
336. 
337. 
338. 


B39, 
340. 
341. 
342. 
343. 
344. 
345. 
346. 
347. 
348. 
349. 
350. 
351. 
352. 
353. 


LIST OF ILLUSTRATIONS Xx1 
PAGE 

An Examp.e or A Frequency Series (So-Cattep) Wuicu Cannot BE 
CVAD ra CN GI ye AC Ei eo oko. 343 
avo oivea ARE Al wavs, POSSIBLE, bec ald 403 0 sack inbphicch aise - 344 
SHowinc THE Four Possiste Cumutations For Point. Dares Mie sasan os 345 
Tue Four Possipre Cumurations For (Pornt-anp-) Pertop Data.... 346 
ir eLorn ir Di OGIV ES yaks ay ee ot ie eye on cheat oe eee 
PyOUUmMON Ob THE OGIVEA(STAIRCASED) 0.1... akace san: Saatidnk cue See. 348 
vO PaION OF THE OCIS: (SMOOTHED) cin ircytids ede ds nas ewe ldo OF 349 
Tue Smmpce Curve anp Irs Two Ocives (STAIRCASED)...........-.- 350 
eCTORG WRipo.S DINGWE Ns On el ee ER re et) seas ie iia ams ae eee 351 
Comparison OF Agso_uTe Data 1s Sometimes DiFFICULT............ S52 
COMPARISON: OF ICELATIVEMOATA OTS, FASY44s 46.5 coe G ore sco a acta oils 353 
DECOMDAR YD AT AWA T UNE MIRIGEHT we anjeenie ebay. fa dosed wevanoe xis Bax, 354 
Tue First Measurre—By Count oF ITEMS..............-ceceeeeees 356 
Tue Seconp Measure—By Count oF UNITS............20eeereeees 357 
[Greater (WN Dog 5) Oye hae en One OA ne we eae Sy eee ene a PTE ay 357 
OMNES RING PETE NEVER CENTAGE Gia shy = ooh a i asts tts cele Fahad ets ee ets 358 
DY ATASLOR STH ES LORENTZ Y OURVERy try. i's Jivi aun oiins oo. ee SE sn 359 
DATARTO UE LOT MEH E LORENZY CURVES oo: buses cet 8 ath ote eee 359 
Mie MORE NT ACIIR Vie earl ne nic tots Sore Fanon eee ath bey see 360 
Waar tar Lorenz. ConvyenLeLes. THE LAYMAN) & «0+. 44 see heen 360 
Wiser LAN ECIU ATR RECA NDI kU Yes ae, oon te Baca Bo este cn ae iaaey ne oc aes 361 

Two Curves or THE SAME Data By Usinec Boru ‘‘More-THAN” AND 
IDE SS-THAN ce UMA DUY ES icirctvele RA. cones ctaderge enti demon 362 
i OmCORVES OR: DIN ERENT 9 LLATIA\.1..ce cscs tsi hens wehireteysta Ae oi eieveta ys 363 
Die OCI CAT IPH ORMIG us URTAN GUAR A roy Shy onsts cosines ones bide tweens tots 364 
MEAD SOC OD AR TUM Gp le Stir pirat lh. auch STM Rig sketsl ooh OG oe meh Cos 374 
MPR E MOET: OCA RELH MSs (Ono erin. (cr rath ays FR emt 7, Ee be RS 375 
Tue RaTeE-oF-CHANGE CuRVE—First METHOD...............0.00000- 388 
Tue RaTeE-oF-CHANGE Curve—SEcoND METHOD.............0200000+ 389 
Tue RaTE-oF-CHANGE. CURVE—ITHIRD METHOD..............0000ece 391 
Goop RATE-oF-CHANGE CHART-FIELDS..... Pete. arene CDR 392 
IARRATE-OF-CHANGE PART-DECKSP ORM sc cyeins den ctte > ae ed alse cieelesventns 394 
Paqir-NE CK ICA TE -OR-CHANGE UDAPERE oa, (invalid sis See a bAsahig Bhs weber 395 
AMES PUTED ECK “AE ROM SU GrODOUU nA nice Hac ob obiea reson Salseetdes 396 
WORE ATH ETO PECIAIN SCALE ATutEH ia) ICIGHT dauicis <) Suis chedust- eines a vslee 398 
One Way TO FIND THE RATE-OF-CHANGE SCALE...........- 0 See eS 400 

ComParIsON OF Series LyinG IN DirFERENT Parts OF CHART, THOUGH 
NOTA EUCTIVATINGHGREATIY 02.22). 55, ous. sis on tutes otiae ee + <> see s 403 
IAMOUNT-OF-CHANGE CURVE “(ABSOLUTE) ssc. 5 eh ce5 os boeing lee etsiels © 404 
AMOUNT-OF-CHANGE CurVE (RELATIVE NUMBERS).............22+005- 405 
ATE-OF-CHANGE (CURVES (ABSOLUTE) si .r.co4- seis weppe etiam hoes as sc 406 
Tue PERCENTAGE-INCREASE-OR-DECREASE RECALIBRATION.........--- 408 
SEVERAL Curves DRAWN ON THE SAME SCALE......-..--20ceeeceees 409 
SEVERATUSCALES INTAL SINGLE OPLIT“-DECK 444) 1.7202 «50 Gis easugiom caress = 410 
Suirtinc Curves To Avoip INSIGNIFICANT CROSSINGS...........+++-: 412 
AM OAMOUNT-OF=CHANGESOHART cit iiss 5 cob doln-da viens yortaets So ciel om viseede = 414 
PRAT REOR-CITANGE ROH ART tes a ds ioe eee ns 415 
LONG-TIME, SERIES OF LCCONOMIC DATA 4.0.4 pin ncierd celg coed ols bese sun a obo 417 
SHORT-TIME SERIES,OF ECONOMIC DATA..... 2.440208 ni ses umes snntcee 418 
AGGARERUL JCOSITIONING, OFATHE: CURVE. \se1 5 00h tus Ge ooh ile see > 420 
CoMPARISON OF RATE-OF-CHANGE AND AMOUNT-OF-CHANGE CuRVESs..... 422 
eR POLATION ANDMEXTRATOUVATION: Gr. eeia cineca: shale hs ages 423 
GER URIT ME SCTR A POLATION! Sete Rahs a neice a ied dae odes slew Be 424 


LIST OF ILLUSTRATIONS 
PAGE 
CompounpD CuRVES..... mL A a, ea eaten a JN ds Be ia teens Rae 
A Frequency Serres WuicH ApPeaRS SLIGHTLY ASYMMETRICAL...... 428 
An ASYMMETRICAL DISTRIBUTION. .....:--01--s+2-0% AE RAR Be Alain AS 
Yute’s ExaMPLe OF A U-sHapeD DIsTRIBUTION......-..-++> Sei AU 
Six MoperaTELY ASYMMETRICAL, DISTRIBUTIONS... ...--00 ++ eeseereee 431 
Tu INDEPENDENT VARIABLE 1s MEASURED FROM AN ARBITRARY ZERO 
POET soc, ec 4 Seg eo PA SPC aeoe, Gore ee eee 432 
A MoperaTELy ASYMMETRICAL DisTRIBUTION WHICH THE LOGARITHMIC 
Scate Has Not Mave EnrireLy SYMMETRICAL. 30 . 434 
Aw ExtreMELY ASYMMETRICAL DisTRIBUTION MapDE Syuernickne BY THE 
LLOGARTUMMIC PROJECTION 05-00 ci ecste ae tt eeadicn oy else wel nener seer 435 
Tue DovusLe-LoGARITHMIC PROJECTION 18 BEsT..............++++-++: 436 
An Extreme_y AsymMetTrRIcaL U-sHarep DisTriBUTION BRouGHT TO A 
Beautirut SYMMETRY BY THE LOG-SCALES.........-.---.++->- 439 
SAME. AS: PREVIOUS—HIISTORICAL COMPARISON. ..-.:2..-.-0- ue dae ar 440 
A TABLE OF THE CONVENIENT, NEARLY-GEOMETRIC INTERVALS BY WHICH 


THe Rance Between SuccesstvE Powers oF Ten May BE 


DIVIDED Oo 60 oe ORAS ee RE ee nen eae 441 
Dire. |-sHARED DISTRIBUTION] 9. o2e cen dein at | ee ee ereeeer 442 
Ocives Prorrep Upon Locariramic Horizontal SCALE.......+----- 445 
An Ocive PLotrep Upon Botu LoGaRITHMIC SCALES.........-.+---> 447 
Tue NorMat-curVE OrpDINATES OF THE UNCUMULATED SERIES........ 452 
Tue Norma Ocive OrpiInaTES OF THE CUMULATED SERIES.......... 455 
Diacram SHowING THE METHOD oF CONSTRUCTING A PROBABILITIES 

PROJECTION ALONG THE! Y AXIS < 905 mie ene ele eee 456 
A Less Userut Form 1n WuicH THE INDEPENDENT OR X-SCALE OF THE 

RancE Is READJUSTED TO STRAIGHTEN OuT THE OGIVE........ 457 
COMMERCIAL: PROBABILITY) HORMS qccteuckmeoeeie no Ocoee 458 
Tue LocaritHmic ProBaBILITIES PRojECTION IN USE.............--- 459 


AN ARITHMETICAL PROJECTION OF THE DEPENDENT (OR FREQUENCY) SCALE 460 


A ProBaBILitigs PrRojECTION OF THE PREVIOUS CHART..... Korea cer 461 
SYMMETRICAL, SO-DAR ASE DATA OBTAINS em ames amie clacton 462 
SYMMETRICAL BUT DisrINCTLy: Nor NORMAL. -.+-.4. 5: oe ooce seen oe 463 
Tue Comparison oF Ocives FoR DIFFERENT DATES..............--- 464 
Tue Octve or THE Units oF MEASUREMENT OF THE Items (STAR BriL- 
LIANCY) IN AN INCOMPLETE SERIES IS STRAIGHTER AND More 
RELIABLE THAN THE OcIve oF THE ITEMs (NUMBER OF STARS).... 466 
ANOTHER EXAMPLE OF THE Two INTERCONVERTIBLE FREQUENCIES FOR THE 
SAME’ DATA icin Ratoe Senet ee 467 
ALTERNATIVE Data YIELDING THE LoRENZ CURVE..............-000: 468 
Tue Dousre-Propasiities PRojJECTION STRAIGHTENS OuT THE LoRENZ 
Curves Wuen or Norma DistrtBuTIons.................--- 469 
DovuBLE-PROBABILITIES PROJECTION FOR SEVERAL Lorenz Curves..... 470 
A Historica, Retrospect with ReverseD Loc PLorrinG FOR THE 
HORIZONTAL, OR INTE WACKIS Se) ee yen creer ee 474 
ANOTHER: EXAMPLE! OF THE ISAME® 4 oe etre ne cee ne nn eee 475 
Tue Four Lower Curves Fatt to StratiGHTEN Out on LocarITHMIC 
VERTICAL; SCALESW iS. ereaooesc coe ee ee 484 


THE SQUARE-ROOT PROJECTION OF THE VERTICAL ScaLE Brincs Mucu 
GREATER REGULARITY TO THE CURVES OF THE PRECEDING CHART 485 
Tue Two Ways oF STRAIGHTENING Out SemI-cycLes OF A SINE CurVE 486 
Suowinc How Ctosety THE Cycies or One Set oF Pertopic Economic 
Data Approacu A SINE Curve WAVE 
Tue Linear Equation, y = ax+e.........5. 


FIG. 


392. 
393. 
394. 


395: 


396. 


SENG 
398. 
599: 


404. 


LIST OF [ELUSTRATIONS XXiil 


PAGE 
THe Curve oF y = ax® oR LOG y = LOG A+-HDLOG HK... eee eee 494 
Tue Curve oF y = ab + € OR, LOG (y —¢) = LoG a -+ b LoG-x......... 496 
Tue Curves or Known Powers, y = ax* + ¢ and, y* = axn+e...... 498 
a C4 
Tue Hypersotic Curves, y = - +c ANDY = paren eee 500 
Tue Hypersota with THREE ConsTANTS, y = a = dike See 502 
PDH EE ARABOLAS Ve =o Ure. bX i 1c Me de accls alcsoyesr ds ope nas. sos ave AS wewtersins 506 
EXPONENTIAL OR: LOGARITHMIC CURVES & 46.5. cg.cthe cossesecsis vis 2 os vcidieus ule © 508 
YS DRAG Chai geet cg paid Sateen ae BO eet Nar Oe en nT” On ipa 512 
Ub VE BDC SRE BER EROS TOR TO OO ee eee $13 
G7 a Oe ata) Ce ae A Ree ayer te cas tir ovat tara cease ol Reveal ca visas ousiisos aay leva Eee 514 
ENG Bee ET NR PL Cie 5, cas ramsgec ths cahag sgh sada citrine EGA $15 
VE LG DGS POR 5 BRED 0 Re PT Te eR Bra Sones A 520 
VO SS BS DIOS Peo eS ee en AE SA ene er | ae eu 521 
Am SIMPLES OAL CULATINGIMCHART 95 cr sccct tcc cere fn oneicteiae clone S22 
Vs 8 NOES So's CE AS OR A OD A DE Fae I Ae 523 
CuarRT FoR DETERMINING SCALES OF CURVE-CHARTS ........-eee0e005 524 
vA 
bh = 5 525 
Y -—3 
Se Ee gee cd a ae oF ear cds ae 526 
Wocuia—nGOSX —- 2) TAN OZ. « vinnie se siete cs dates esasceews care we 526 
z= (4 +2):+ JB eae Nae at» ea UME ACRE an ois SIN 2 ee 527 
Na OREM COMPIICATED: CHART ce niet aia ose a asunectors tele Gane eins orteete 528 
A Stmpce Mopet oF PROFIT-AND-LOSS COMPUTER............2--.000- 529 
AE@OMPOSITEL CHART With. NIANY SCALES 9... 54.0604 ase cleuee le sah $30 
A StmpLeE CoMBINATION OF LOGARITHMIC AND ARITHMETICAL SCALES BY 
THE UISEMOF -ACOURVE wie oc cried cae CEI SRR id Ay etna RE aate 531 
AMS IME UES ARAL IE UMINOMOGRA PHihcc 5. pate 2 eicutacss, cite atereelarevsasie saree oieeesol ates 534 
Re i ee ere ee Pc octet sa nee ae caso aa EP 535 
TF ce DE AS IB © OR are ts ea RIEL at te ee eR ee 536 
VB AGG 2. & eR OO EO OE ATES Rea bracts are ener 537 
Nae Oe a ea oo ee oc a. iss opateuelsto yj aherw oie A eae ate 539 
FP Se BC ASIA, SR SS Rae OTC ete Ee Oe RE On enn Tan Cer 540 
YE ey ope IE DCs Goode Pa he Bits Sc pi ROAR a Rs OI er oR I 540 
EEN VPICRE DONS CATE Mee tere mie, Garren ope SoU CASEY Ga yuaticucatne 542 
Tue Use oF AN OUTER SCALE FOR THE UNKNOWN VaRIABLE I8 Not Goop 543 
Pe ee ae aoa Sajuibys cad, san Melosh ene ea eis 544 


CONSTRUCTION OF THE PARALLEL Nomocrapu—l.................... 547 
CoNSTRUCTION OF THE PARALLEL NomoGrRAPpH—Il]................... 550 
CoNSTRUCTION OF THE PARALLEL NomocrapH—IlI].................. 552 
ConsTRUCTION OF THE FacroriaL PARALLEL NomocrapuH..........-. 555 


Guar LOR] DE LERMINING SIZELOBL IY PE. «salience snroaies sca ereketie 556 
CHART FOR DETERMINING SCALES OF CURVE-CHARTS........0...00008 557 
PARALLEL NomocrapH Not CHARTABLE BY FORMULA..........-..+++ 558 
Ta Pe ATS ce Rom iar tRatRT Sas ROT XS aes as dares eel sols 561 
SHED YESCALE OUTSIDELTHES PARALLEL GSCALES tm is Wels viele wei caine. 568 
IMIEEVESCATE. INSIDE (THE. CARAULE DE SCALE GUE, ear htiney tu. Fe se, 569 


ConstRucTION OF FacrortaL Ziczac NomoGRaPH—UNnFINISHED....... 570 
ConsTRUCTION OF FacrortaL Z1GZAG NomoGRaPH—FINIsHED....,.... 572 


LIST OF ILLUSTRATIONS 
PAGE 
Cuart To Construct PARALLEL NOMOGRAPHS......scssceceseoesees 573 
Cuart To Construct Z1GZAG NOMOGRAPHS........ceeseseeeeeseeeee 574 
In Quapratic AND Cusic EQuaTIONs THE PosITION OF THE CENTRAL 


Axes Becomes VARIABLE, AND A CHART-FIELD TAKES THE PLACE OF 


RUSINGE EM SCALE see oe Ae Oe ae av ee oe a oe Ra 576 
AOS TATTONARYVORS FIRED | ULE steele toel ates toate clever tebe eintba tenet 578 
A STADE RUDE od «2G er uik cope Se ey See ae E oacaegin oe ait ee eee 579 
Tue Macniriers INCREASE THE AccURACY OF READINGS............ $79 
SlIpE: RULESWITT CDHRER: SLIDES seudere eo ecate ls seats es ein reo eres 580 
AVIGCIRCULAR SLIDER RUDE POCKET SIZED rs at clans © 9 oekte es aetee tetas 581 
AS SPECIAT CIRCULAR SLIDESR ULE + ta aco eae a nalice tater lee oeries 582 
AY CIRCULARS SLIDE RULEOWITHEMIANY uVIARTABLES'. . .escie cot anne 584 
Tue Same AS THE PRECEDING, Except THAT ALL SCALES ARE COVERED 

AND SEEN OnLY THROUGH SMALL Open SLots or WINDOWS..... 585 


An ARRANGEMENT OF PULLEYS, WHEELS, AND WEIGHTS, BY MEANS OF 
Wuicu THE Pornters Come to Rest aT THE Roots OF THE 


POUATION ts fo teconteead ane ee Sones oe Retest ee ee 586 
mh Gee PO ea AS ete a SIS OE pale ord | a ey oR A re oo oe 588 
4 RE ae hte A Lt UR rehe ty Ptah Ra ein rn eo URNS Ac: un cee 590 
Tue HunpReED-PER-CENT TRIANGLE FOR Foop VALUES..............- 591 
Ep PAP RRA io Sect ater ah Ah agp ANN ope Pa cibhe CaaS RVAOS ome hie oe RE Oe 592 
mes BACTORTAT.. LOOU, CURTANGUB cic cattery soem eieehaius etaeieeh hier ee 593 
ASSINGLE ASCALE AU SED SEOR HLWO PAKBES. «cas temcnie oc ciekeretateieva sate erecnian ine 594 
ee Rey oR EN ak * SOTA wey eee Rae Seed OD Oe Te eee 594 
DR he PI eer nak Lor nd avte Mach com te ic Se, ae nar Tae ate RON RE 595 
Rs cfc ATM, eeruel thier caudate ye Rea eae bate eS On eae ee 596 
BO ae ME ete Tees 2) ech sel Sah ATV CREAAI RE ie aeaa AC ae Pee RRR ARE RTI oe 596 
SDHB se ORIGINAT: DATAWFORW AY IOUSG SOUARE ce (aera alee ereae 600 
INU HISsEORMATHESDATAsIS NOT: GHARTABLE a0 ites ose che eetanee 600 
HERE: WACHS ROW SlOTAES MOOG, a. cc core omnes oie racic creer eee 601 
HERnH ACH. COLUMN TOTALS! hUOG Ame aur arhmnte Sane ace 6 acer ee 602 
Tue Primary Diviston ALONE PLOTTED FROM Fic, 462.............. 603 
iE COMPEETED SOUARBovauned tore ian wrote errcee anes ane een eee 604 
Here THE Primary Division 1s THE HorizontaL Ong, PLorreD FROM 
| GIKCHGe. Cobh ete tape ee Pee PRN Cane Rieti e eRe MEE Pl Be 605 
AMOO Ue RECTANGLE BX xan tre ca pccaterenatt ies tare ater eo) epee rn ee 606 
ATALOOUG SQUARE hechk sce Ree or ciclote thet cites Otee ene eee 607 
ANOTH ERD L009G SO UAR Boas Gare a onhea tok Card aie tn Rr eA 608 
SAMESAS THE GWAST: IN) CIRCULARS ORM Aca iirc incre oe eee 609 
ANOTHER: OOS Cinch By. fyes mens orecta uit arent tet ert ce eae Rene 610 
A Turrp CLassIFICATION HAS BEEN ADDED Here By Diaconat Divisions 
AND) SHADINGS) SHOWING SEX avai pacraaern ee eae eee 611 
A SIMPLE AND ExceLLENT AREA BAR-CHART.........i.eeeeecccsees 615 
VERTICAL AREA=BAR Siu colt atone ete ae eee eet ee a ne 616 
COMPOUND CAREASBARG  c, cu trcet emir eae ee ee ne 618 
BALANCE OR CouNTER-POISE CHART WITH Two Facrors............- 620 
A More Picroriat Form of THE PRECEDING CHART......-...+ee++eee: 621 
Every Map ts aN Area Cuart. On THIs Map tHe Areas REPRESENT 
Square Mimiws.nc,2 Ser cee tc cee eee eae ee ee 624 
Due Usuais Mar or mip UNITED STATES |e eee ae nee 626 
On tTH1s Map THE AREAS REPRESENT INHABITANTS..........-e-eeeeee 627 
APLASTER-OF-PARTS: VIODET MEN Hote ee ee ne eee 634 
A> Conrapst ace Opera eee run sate OA een ee eer ane Bae 636 


‘SMALL SratistrcaL Departament. be ae 


st ben IPENGAS ior te AA ee PRE Coen etic ease 


Ver SU ere Bee ‘=e - 


in 
\ 
\ 


JAR tO ee 


en Sa ; 
a eater ; 9s 
+e 


iy -_ 
ns i NES 
ee A ae 
: 


. 


ie 


> ~ 


LIST OF ILLUSTRATIONS 


By SuBJECT-MATTER 


FIG. PAGE 
1. Pictorial Map: Natural Resources of the United States. By Mr. C. Vande 
ACTING #0. es cn nc a EEE ERE cet ES a EP Tp Pre AE RAPA 2 
2. Heart-shaped Map of the World. From Bartholomew's Atlas........000. 3 
3. Mercator’s Projection of the World. From Rand, McNally & Co........ 4 
4. Hemispherical Projection of the World. From Rand, McNally &§ Co..... 4 
Dee lpticalserojection ofthe Worlds nes nents Gl. oe tie aor et oe ae 5 
6. Homolographic Projection of the World. Adapted from maps of the Ham= 
LONGING DIC OPIS ER ra err et ey nee nee teen rato) 
Sampoamples/of Cross-ruled baperss vu cine dice aude os Sec c evens teens 7 
LOseh loot-planvota omalliscatistical Offices). 4. wore ees acer cole cee oucstes 10 
iipeline INomenclature;ot-Co-ordinatess: - + fie eet eee deen eece 11 
12. A Classification of the Non-mathematical Charts................eee00 14 
13. The Structure of the League of Nations. By Mr. Sidney Gulick.......... 16 
14. The Structure of a Large Merchandising Organization.................. 17 
15. Joint Interests of the Big Five Packers. From the Federal Trade Commission 18 
16. Interlocking Interests of the Packers. Data from the Federal Trade Com- 
CLEA TOT ho it ol Bice Ue ae pee Nee N Ses Cetin ecaece NOR Po eer DN A PEER IE SI 19 
17. The Evolution of Animal Life. From Thompson’s “Outline of Science”’.... 20 
1S) pemotapesiumetne Making of Curve-charts...)ocess 0. 6s eee wane nue. 22-25 
22. Density of Population of the United States; rank of the States at census 
EADS ST 192 Om ea Ries aan er ols coh tees Rh eka Hee Neneh ae ReaD eae 26 
23. Chess Openings: The Evans Gambit and its immediate alternatives..... 27 
24. Diagrammatic Logic of the Gantt Charts. By Mr. Walter N. Polakov.... 28 
25. Routing and Channels of Sales Efforts. By Mr. Richard Webster. ....... 29 
26. Flow of Supplies in the American Expeditionary Force. By Mr. Malcolm 
CRROTL Se I eae See aha heating Mera ee nike aa ners 30 
27. The Round Flow of Money: income and expenditure. By Mr. Malcolm 
(OL, TAR Cests who bess Sep eed wi teaah certo Crete rohnert te ar 31 
28. Process-chart of the Loading of VB Rifle Grenades. By F. B. and L. M. 
(MEAS Sec pe SS RRO OTTO ESOS OE OO OR CE erg ety EEE 32 
31. An Analysis of the Stock Inventory. From Clark’s “The Gantt Chart.”....- 35 
32. Scheduling 8-hour Turns with 24 Hours Off after Six Days. From the 
SS ETE LUNO [EC CUOPESLGIUSULGS Re ror ees rue peat See os gh vaca shales oY pegsicas APE (set 36 
33. The Weekly Cycle of Sales in Department Stores................0-0005 37, 
34. The Annual Cycle of Sales in Department Stores..................-.0% 38 
35. The Salesman’s Routing on a String-map. From Rand, McNally & Co.... 40 
36. The Analysis of Sales at Local Branches, by map-tacks. From Rand, 
AV CENCUIBD OM Oaks os abn Se at LORE CRO ONG Pac Bae Care eR rac era Mic ReNanstet 41 
Same leoutine thea orkconiasstatisticalpeportam. ttre entities s+ 44 
38. Flow of Goods and Money through a Large Merchandising Organisation.. 45 
39. Map of Garden-planting Times in the United States................04. 46 
40. Distribution of Metal Money in the World; approximate stocks in the chief 
COUNETIES MLO MG me tied tate etme eee Sts ae rate Aine ay chal ve tei wumee Sos, gan ahe 54 
_ 41. State-groupings used by the Census, the Red Cross, and the Audit Bureau 
GHGirenlationsse rete or ee ae miriam yo renner aa@iee 56 
42, Pig-iron Production in the United States by States, 1920. .ssrveeeveeeee 57 


XXVIl 


XXVH1 LIST OF ILLUSTRATIONS 


FIG. PAGE 
43. Illiteracy in the United States; illiterate percentage of each class (by age, 
sex, race) for each group of States, 1920) EE ieee iS cio eter 

44, Illiteracy by Age, Sex, and Race, in the United States, VG20 Sis Site oretes 60 


45. Savings Bank Statistics; number of banks and depositors; total, average, 
and per-capita deposits; and ratios between banks, depositors, and popu- 


lation» United States 1820=1920) m= mectnes «cee eocne near tokens 61 
46-56. The Evolution of Cartesian Co-ordinates... . 12.0.0... cess ceeeeeees 63-72 
G7 Polar GovOndinateSs <n; cies Sateen ake re corel aie Stade sence tio ete rolarel stahstmieberepetene 72 
58-65. The Principles of Linearsillustration\. (gir elon sae.) eee 78-81 
66. Foreign Trade of the United States, 1920, Divided as to Exports and 

Timports 3s as shade tle So ene ahs halen he ieee Meee epee 83 
67. Periodicals in the United States, 1920, Divided as to Period of Issue. Data 

tronteNaW Aye 5S Sone aac + tapes Glee oer deci tenite sale aaerany rake 84 
68. American Casualties in the World War, 1917-1918, Divided as to Cause and 

IN ACUTE scroibct stein eves dou ei eacas soa onl Uae oes s dN SIRNO Rn ea ae ey eee 85 
69. -The Family Budget, Divided as to Classes of Commodities, U. S., 1913. 

Data from U.S. Burcau of Labor Statistics y.reioras te ost eel 86 
70. Foreign Trade Gateways: ports of export and import, by specified groups. 

Ws Se D920 enc eipreia tly GEE. heaters Sie Saale eres Cen RP eee ee Saree: 86 
71. The Family Budget, Divided as to Classes of Commodities, U. S., 1913, 

1920, and 1921. Data from Bureau of Labor Statistics..............-- 87 
72. Imports into Russia, 1921. Data from Russian Information and Review, 

TEGUG ONES, case ecs re oe BR FOL RE Ne ih OEE A Ye 90 


73. World Statistics: land and sea areas; continental areas; geography of the 
land; population of human races; language populations; religion popula- 


tions; continental populations; continental languages................. 91 
74. Purchasing Power of the Dollar of 1913 when Used for Food at Retail, U.S. 92 
75. Retail Food Establishments, New York'City, I921> 07... ..ccc<e 93 


76. Analysis of Cost of Shoes, Shirts, and Suits of Clothing, as to Raw Material, 
Labor, and Overhead Costs and Profits, of Textile Mill or Tanner, Manu- 
facturer and Retailer. From the Federal Reserve Bank of New York.... 94 

77. Analysis of Expenses of Retail Stores; expenses classified for department, 
shoe, clothing, hardware, grocery, furniture, jewelry, and drug stores. 


Data from Harvard Bureau of Business Research... 0.00. c cece eee 95 
_ 78. How to Calibrate the Circle for a Scale of Percentages... .. Se ae BOG 
_ 79. Cost of the World War to the United States, as of July, 1921. “Dorn from 

the World: Almianacas® cares es Gon shone ok Le ee OF 


80. The “Swift Dollar’; analysis of income from sales. From Swift 3 Co..... 97 
81. Business Failures, United States, 1920; amounts divided as to nature of 


Business ey. eS 5 eae tas feo eT Toe ELT OTST eee ere 99 

82. Density of Population of the Earth, by Continents. .................0e- 100 
83. Presidential Campaign Expenditures, U. S., 1920. Data from Senatorial 

Commitee one oer eae hate OO e nee eee aiotetovalelevels arersien LOM 


84. Farm Property, by States and State-groups, in the U. S2 1920, in Vatact . 102 
85. Trade Union Membership of the World, by Countries, 1919. Data from the 


International Labor Office ..<...c Hn ve ER Oh Meee a ee 103 
86. Per-capita Public Debt (less cash in treasury), U. S., 1800-1920, by census 

YOATS 6 osc ahs access yack barge eeu Somes So aera laste RSE ve ee 104 
87. The Causes of Fires, U. S., 1915-19; value destroyed by specified causes. 

Data from the National Board of Fire Underwriters, N. Yo... ..0..0.00e 105 
88, Religious Denominations in the United States, 1919. Data from “Year Book 

Of Churches” accae sere 4 dus. 4 5 MOORE he ete Tee NEE ee tOGam 
89. Analysis of Soft Coal Production in Indiana, Illinois, and Ohio, 1917. 

From Mr. Walter N. Polakov 


FIG, 
90. 


OF: 
S35 
94. 
o5. 
96. 
OF. 
98. 
oo: 


100. 
101. 


102. 
103. 
104. 
105. 
106. 
107. 
108. 


109. 
110. 


1 


112. 
113" 


114. 
115. 


116. 
117. 


118. 
119. 


120. 
eile 


LIST OF ILLUSTRATIONS XXiX 


PAGE 


yee Foreign-born White Population Divided as to Country of Origin, U. S., 
Average Weekly Earnings in U. S., Aig Cost of Living. From Federal 
RESET EAM Oe OPIN COE OTIA Ia re We coin Sea Dene ee hws 
Accident, Frequency, and Severity Rates in American Industries. Data 
POmDUReaIb OL Labor Sialtstics 0.50 Oe Re a 
Fatal Accident Rate per 1000 Workers in Coal Mining, Specified Countries, 
V9” Data from: Burcau of Labor Statistics. 000! oo cc. coors en ve ee 
Business Failures, Amount of Liabilities, U. S., 1916-1920. Data from U.S. 
(COPA ex chee cng dl epasn sii Ub Ge EEO OE Ht RE Aa Be eae eae 
Direct Cost of Great War, National Debts of Chief Belligerents in 1919. 
Data from E. M. Friedman, “International Finance.” .........0..00+05 
Coal Reserves (Unmined) of the World, Millions of Tons, 1920 Estimates. . 
Pre-War Occupations of Wage-earners in Seven European Nations....... 
Occupations of Wage-earning Population, 10 Years Old and Over, U. S., 
NOLO AA TOM UR SMOCT SUS hem ye weeds ee eT Sate 
Foreign Trade of the U. S., Classified by Nature of Articles, 1920........ 
Publication of New Books; in Leading Nations, 1919-20. Data from “Le 
DJVOLH RAMI CU TM ATUS te Pe. Te en SE es 
Ratio of Gold Reserves of Central Banks to Paper Currency in Circulation 
Compared with Relation of Exchange Rates to Par Value (March, 1922). 


Prom Ecderal Reserve Bank of New Yorkc..)2-0:.5-.40 422+ 0.20- eens: 116 
Total Number of Immigrants Arrived in U. S., 1860-1920.............. 117 
Countries of Last Permanent Residence of Immigrants Arrived in U. S., 

TUS OE CHAD)» ceacteaes Meme l eh ore ea tiasa nce einem | re Simin i CAEN, He Wea pr ROR Rae 118 
Class Alignments of Population, U. S., 1870-1910. From data derived from 

CENSUSHO VIA MLE LL GHSC Tse he rete oe eee a a ce reat nie Me ea es 119 
Percentage of Imports Received from, and Percentage of Exports shipped | 

to, Different Continents. From Federal Reserve Bank of New York..... 120 
Books Published in U. S. and England, 1920, Compared as to Subject. 

Daayronj Lie pabushers: Weekly. Ns Von aoe oe ene associa ese ape se » 121 
Sex of Emigrants and Immigrants, U. S., 1917-1920. Data from Report of 

U.S. Commissioner General of Immigration... .. 12.06 cece eee eens 122 
Wrbanwbopulatiomomlmonel oo 0ecrtorcweecrseestcie nacre oe em iakre: 122 
Combined Exports and Imports of Leading Nations of World at Par of 

[BS eR 4 wineries Oh ee tat A ea he tn a SCL ten aca ea eeu ee a Pe 123 
Highest Prices of Food at Retail Gndex numbers, U.S.). Data from Bureau 

CAL OL SI ALDS ICS Mien GRA Uae Readies is hae Ae Wea Seen om, ays ae 126 
Rorerenwbradeopthe Worlds byi@ountricsine sss ee. ocean * aaiace 127 
The Ideal Philanthropic Budget, U.S., 1921. Data from Paul and Dorothy 

| Dynes, VINE CODE IGE: Hi CHEE ROACH Gn OG dae vin aon on Aca 127 
A Half-Century of Progress in the U. S., 1870-1920. By Mr. C. Van de 

HED ey 28 & ej CRORE REELS POSS OOD OST 128 
Automobile Production, U. S., 1913-1921. Data from Natl Automobile 

eA OP COGIMAGR, Bio 6 NOOSE lt SB EDUC DE HAG O90 06 UEe NOdD Bae 129 
Savings of the World; per capita deposits by countries................4. 130 
Production of Basic Commodities, U. S., 1922. Data from Federal Reserve 

TEPER he eis ALIEN BS ET OTN OE TC OOO TUR ORT OEE 131 
The High Cost of Living, U.S., June, 1920. Data from the “Monthly Labor 

PRS RD, dope hd SPORES HERIOT OSE TES Se OE AT OC tae 132 
Production in the United States, by States, 119........... ccc eee eee ees 134 
GoldeReserves ofthe Worlds 191s to 192 len meine sis sete «es wield 135 


U. S. Production of Specified Commodities Compared with that of the 
VU oye Ce: ak diye, Setoreecab Seen oe DSP OOO GCSE ST Oe NCLE RCRE Eon 


XX LIST OF ILLUSTRATIONS 


FIG. PAGE 
122, Fatal Industrial Accident Rates for Specified Industries, U. S., 1913. Data 
from Bureau of Labor Statistics... 2.00 c cere enter eee cere ecasocves 137 
123. Death-rates in Warfare, Shown as to Cause, War, and Country. Data from 
Offical Uc SP RUUCLIM = sek ee tee, eae ee eR ee ee See lss 
124. Accident Mortality, by Age and Sex: U8.,0910-12 cot etn aren 199) 


125. Male Accident Mortality Rates, Shown by Vee and Nature of Accident... 140 
126. Foreign Financing in the U. S. and the United Kingdom. From the Federal 


IRescroe cBanenOy Nik oe cen eR ala hee eee eh eka kee eliaeiety 141 
127. The Blame for Industrial Waste in Specified Industries. Data from “The 
la mimat1On: Of WAGSICS! ome coe ais Se ctase: We Seo ov aA to Lale oe ed . 142 
128. The Nature of Industrial Accidents, New York State, 1911-13. Data from 
ING Vic Stale De ptxOf Lavo tyers carey areiatty ois eioiaiaiasiereieteeies See Wein 143 
129. Gross Tonnage of World Seagoing Iron aaa Steel Ships, 1914 and 1921, 
From the Bederal Reserve: Bawkiof NeW asa: sepia depic eieineyladciimiehie cave 144 
130=S:m Populations Ui9s55.1 7901 920 gr te Mens ck cab ogercaepenar ate caetoatsieey treet 145-8 
136-7. Imports into Rissa, 1921. Datafrom ‘Russian Information and Review,” 
LORD ORS es sea fei cesses eels ON: ae eres ate cunuitesand 150-1 
138-9. The Amputated Chart. hon Me John Wenzel es an tics ietreirtenete 154-5 
140. Curative Effect of Diphtheria Anti-toxin. Data from U.S. Public Health 
OY is ie) OE MONE SW SEE AS SRN MER io SERRE ICR Cor roca. Mae OS 156 
141. Workers’ Output and Fatigue, in Dexterous Hand-work. Data from UTeiSs 
Public Eealil: SEroice 2 xc oles gah caus as tani Wich «eit eel or eect eT 158 
142. Average Prices of Liberty and Corporation Bonds and British War-loans. 
From the Federal Reserve Bank of New York.........0.00e-scvecessene 159 
143. Adjusted Index of the Volume of Manufacture. From the Harvard Bureau 
Of ECONOMIC ERESEGTCN sessed vg hus fs olor AA gece els Rl a os Sete ETS 159 
144. Income of Railroads. U.S., 1920-1. Data from Interstate Coma Com: 
TILEY Oo RR ee RE EN rect EERE NG om Pee e bc 6 160 
145. Production of Automobiles, U. S., 1913-21. Data from Nat'l Automobile 
Chamber of. Commerce's. orpces eesti ied eckson ES 162 
146. Comparison of Prices of 14 Basic Commodities during Civil War and World 
War. From the Federal Reserve Bank of N. Y........ <eisier anc ete 163 
147-53. Effects of Changes in Curve-chart scales...............-..+-+-.167-171 
154-60. Samples of Suitable Curve-chart Plotting-paper...............000 173-9 
161. Capital of New Incorporations. U.S., 1918-20. Data from N.Y. Journal 
Of Commence s Snar,, « «piety Pe 1G ce eS arse costo Anica ROE A ere ee 180 


162. Fire Losses, U. S., 1875-1920. Data from the N. Y. Journal of Commecee = Loe 
163-4. United Cigar core Co. Sales. 1921. Data from the Survey of Current 


BUSINESS. ie waiter tiapsce cree ae esha CLE SOE EEE a eC eRe 182-3 
167. Samples of Chart-paper Published. Fen Mr, Jolin Wenzel: tas steers 186 
174. Food prices in France, Great Britain, and the United States, 1920. Data 

Grom the Monthly Labor Remsen. etd. tr vie) a eer ee 194 
175. Trade-Union Membership of the World, 1910-19. Data from ie Taur 

mattonal Labor Opjice noel. tes Wayans Soe sees i cee ae eee 196 
176. Oil Consumption, U. S., 1911-1930. From Joseph E. Pogue’s Er naa 

I add) RR OES Hh GOUT EOIN ore. G Sate Cpa sake 197 
177. Production of Automobiles, U. S., 1913-21. Data from National Msamana: 

Ghamber/of COMMENCE.« cccmsc pean On NO Wl ee RTO 198 
178. Invention and War; comparison of patents during Civil War and World 

IW ia ca iescssynisiviava ic les ci she SSI TO ANCENE ca trata ei re evento trois ecto een ee 199 
179. Employment in the U. S. and N. Y. State, 1915-21. From the Federal 

Reserie Ban kof Nis «aici tok reed eee, A ee eR ee 200 


180. Prices and Volume oe Sales of Stocks and Bonds in N. Y. Market. From the 
AUNGLISE. ois eResre 4 oats nee aon FA CE ee Et 201 


‘LIST OF ILLUSTRATIONS XXX 


FIG. PAGE 


181. High, Low, and Average Rates for Commercial Paper, 1831-1920. From 
epopncortah Neversebate of Ns YF. i. oe sl eos dh esi a 0 202 

182. Retail Food Prices. U. S., 1919-21. Data from the Bureau of Labor 
SAID UGG ch us eons nm take Ain te tae are Neat ge Sree Bee hse 203 

183. Stock Prices and the Call Money Rate, 1914-22. From the Standard 
: SEPT ORAGES (Bas oc OR APE st gh Toe TN rg 1 NCR a ah eer eR 204 
184. Cost per pound of electrical machinery. From Leonard A. Doggctt....... 205 
185. Magazine Advertising, 1913-21. Data from Printer’s Ink... ............ 206 

186. Size of the American Expeditionary Forces and Armies in the U. S., 1917-19. 
HIOUVig aL CONAGRA TES MAREN CANO te ch oo et eg 207 
187. Magazine Advertising, 1913-21. Data from Printer’s Ink............... 208 

188. Common Labor Wages for 10 Hours Work, U. S. Steel Corp. From Mr. 
bE TODES Net tc toto Me RoR OR EA Eo a Cape 209 


189. Open Market Interest Rates and Discount Rates of the Federal Reserve 
Bank of New York, 1921. From the Federal Reserve Bank of New York. 210 
190. Call Loan Renewal Rate and Prime 90-day Banker’s Acceptances, at New 
York. From the Federal Reserve Bank of New York.................. 211 
191. Bond Sales, 1889-1922. From Mr. Leonard Ayres... 00.000. c cee 211 
192. The Family Budget, U.S., 1914-21. Data from the Monthly Labor Review. 212 
193. French Women-workers during the War, 1914-20. Data from the Monthly 


LAA GE] OTIS SOE CE he LOT ee ee ee 213 
194. Class Alignments of the Population, U. S., 1870-1910. Data from A. H. 
CISC MRD Oe tel oA top SN tee HE PETE ER, Ca aR ea ae 6 214 
Poe) ne Nature. of Export Goods, U..S:,.1910-19 5 oi ee eee ee ne nb 215 
196. Imports into the U. S., by Country of Origin, 1800-1920................. 216 
197. Exports from the U. S., by Continent of Origin, 1800-1920............. PG 
198-9. Consumption of Gasolene by Classes of Uses. From Joseph E. Pogue's 
Se COMOMTGSROIMIRCLTOLAU TIT tata itis. une uate edd ane eacael As A OI» 218 
Pm @hancessincthe stand and: Of Livin g@svc. mae: «cect ns Sy oA gee cree 2 ae 219 
201-12. Capital Invested in New Incorporations, U. S., 1919-21. Data from 
NESE) OUTIL OP NAOT IUEN CO mee Tesi ecco aserg) sthceie oes, oP Alan aeiolg operant Seer 220-34 
207. Wholesale Price of Bessemer Pig-iron, 1920-2). Data from Bureau of Labor 
SHEDS ENGI oe yo Ek POR PR a RO eto ON ee ER geo iE A RSH es, Re 227, 
213-15. Seasonal Fluctuation in Building Operations, 1910-20. Datafrom F. W. 
IDEN OTN Oa Oe SNOT ear d kel Git es Marte teeter = yc oe, OL San rane Oscar Oe 235-7 
214. A Mechanical Steam-pressure Record. From Walter N. Polakov......... 236 
216. Retail Prices of Eggs: 1913-21. Data from the Bureau of Labor Statistics. 238 
217. The Forces of the Business Cycle. From Malcolm C. Rorty............. 239 
218. Seasonal Virulence of Scarlet Fever, Data from the U. S. Public Health 
Sapatgee ooo et BIEN igh ee ee Sn ied Sen rn hs ey Ue eae earn ee Sti Ree 241 
219. Accidents in Manufacturing, I!inois, 1910-12. Data from U.S. Bureau of 
Labor Statistics..3.......- eeu cs RL eg or ont ara EI 242 
220. Cold Storage Holdings of Eggs, U. S., 1916-1921. Data from Survey of 
CUETO IED LS TESS Re Re eT eT eee IT EP TSM acs dae te lbsatas 243 
221. Strikes and Lockouts, U. S., 1916-21. Data from U. S. Bureau of Labor 
SHAS BGS 5 6 ee SO BO OEE ICT encore ita... CoH et ero a egg 244 


222. Egg Production, U. S., 1920-21. Data from Survey of Current Business... 245 
223-8. Capital Invested in new Incorporations. Data from N. Y. Journal of 


OGIO CO ete Ree tT ae ee cetaa cee ereitgpard cee tlever’ 247-54 
225. Typical Seasonal Changes in Interest Rates between 1890-1908 and 1917- 

21. From the Federal Reserve Bank of N. Y.... 2.00002 cece cece es 248 
227-32. Examples of the Zee-Chart. From Mr. Arthur R. Burnett.......... 253-60 


Page em Details onthe Gantt) brogressiChartiae itr sae acer ctiecee core « 264-269 


XXXil LIST OF ILLUSTRATIONS 


FIG. PAGE 


239-42, Examples of the Gantt Progress Chart. From Mr. Wallace Clark’s “The ; 
Gani Charbel ee oe ee ea 270-5 
243 The-Flow of Goods in am Industrysees >t a tee es oat eee rete rater 278 
244, The Flow of Goods and Orders in an Individual Business Concern...:.... 279 
245. The Accumulated Trade Balance in the United States, 1800-1920......... 281 
247, Prices and Volume of Sales of Stocks and the Call!-loan Rate. From the 
Rederap Reserve DAMe Of Nand iGreen ner ne oe eee ener 284 


248-50. Gasoline Stocks, U.S., 1920-21. Data from U.S. Bureau of Mines. .. 285-8 
249. Course of Production in Specified Industries, 1919-22. From the Survey 

Of CUTTENT-BUBUNESS vgs estado anit Ba ach geee ee en ee Eres 286 
251. Commodity Stocks, U.S. 1919-22. Data from the Survey of Current Business 289 
252. Retail Prices of Specified Commodities, 1917-21. Data from Bureau of 


Labor Statistignyc cots ewe cea he eo Se LT) Ce es 290 
253. Production of Basic Commodities, in March, 1922. From the Federal 
UROSCPOE BANE LOL ANe WY sek vente He kn ae aon ere ae ee ee ees 291 
254. Wholesale Prices of Specified Commodities in March, 1922. From the 
Supe) Of \GUPTEIE USELESS Crh, eee acetate 0 tae a ee ea rene 292 
255. Department-store Sales and Chain-store Sales, 1919-21. From the Federal 
WRESE70E BONE OF NGAL gah ck Rie Cee cata a, eee ta eee ete ee 296 
256. Department and Apparel, Chain and Mail order Store Sales, 1919-21. 
Prom the Federal Reserve Bank of Nee. wos ee oe ee 297 
257. Production of Manufactured Goods. Data from Mr. E. E. Day......... 298 
258. Wholesale-price Indices of 20 Basic Commodities, and Dept. of Labor 
Index. From the Federal Reserve Bank of N. Y.........-.0 00 ee seeees 299 


259. Prices of Oil Stock and Petroleum. Data from Mr. Joseph E. Pogue...... 300 
260. Wages and War; comparison of wages in Civil and World Wars. Data from 


Monthly Labor Revirw rnc. | on ie hat eee ee Ee 301 
261. Wholesale Commodity Prices in England and U. S., 1790-1920. From the 
Lederal Revetee: Bank Of Nave. oie eek ee ee te ee ee 302 
262. Wages, Prices and Employment, U. S., 1915-21. Data from the Monthly 
EQDOTER EOE I. lO AS ce a Re a 303 
263. Liability of Failures in U. S., compared with Wholesale Commodity Prices. 
Promithe PederalaReserve Bate of Nd osecn eae ee ee eee 304 
264. Foreign Exchange Rates and Commodity Prices in Specified Countries. 
Prom the Hederal ‘Reseroe Bank of Nia. 0400. 8e oo ee 305 
265. Wholesale Commodity Prices in Foreign Countries, 1915-22. From the 
Rederal Resevve Bawk ofa: Vinee tet Oe Se ee 306 
266. City Finances; Per-capita revenue and receipts by sizes of cities.......... 308 
267. Production of Red Salmon in Alaska by Size of Containers. Data from 
USS. Bureau ofsF isherterice ngs aceon an geehae wae ee 309 
268. Rents in Denmark by Number of Rooms. Data from Monthly Labor 
TREO EtU neces ee eT ROT RS TET NR Ho ee 8 OT 309 


269. Effects of Diphtheria Antitoxin. Data from U.S. Public Health Service... 310 
270, 272, 277, 289, 296, 308, 309. Per-capita Fire Losses, 1919. Data from the 


Nat'l Board of Fire Underwriters............ 311, 313, 319, 330, 336, 348, 349 
271, 300. Output of Workers. Data from P. S. Florence.............2005 312, 340 
273, 274, 275, 276, 294. College Professors’ Salaries. Data from the U.S. Bureau 

Of GULCH ON rior ooh OT EEE Te SE ORE SILAS SU Se eSiL 6.03 Mienoa4, 
278, 283. Duration of Strikes. Data from Monthly Labor Review.......... 320, 324 
279; 282, 301, 302, 304. Size of Farms, U. S., 1920.......... 321, 323, 341, 342, 344 
280, 281. Gold Production of the World, 1493-1919..................... SA lp Soe) 
284. Membership of Strikes. Data from Monthly Labor Review............... 327 
Leo msize of Factories; Uc SalGla coro a te ee 328 


286. Value of Manufactured Products, U. S.,1914................... 328, 356-364 


LIST OF ILLUSTRATIONS XXXiii 


FIG, FIG, 


£87, 306, 307. \ Hours of Labor; U. $4 1914.0.c00cecececes ests . 329, 346, 347 
oP onmuconomical Speeds Of LEUCKS 20 Suc 6a ccs Sh caw be Faves stucdé overs. 329 
290, 305, 310. Size of Families, British Peerage. Dae from Yule’s “Theory of 
SLB pA ater 2” We arly he ae So ee ee 331, 345, 350 
291. Scallop-shells Distributed as to Number of Ridges. From C. B. Davenport. 331 
292. Effect of Tuberculosis upon length of life. Data from L. I. Dublin....... 332 
293.. Bank Salaries, N. Y. City, 1919. Data from the Federal Reserve Bulletin... 333 
295. Stature and Weight of Children. Data from the Children’s Bureau.. 335 
297. Workmen’s Compensation Payment Delays, N. Y., Pa., and Mass. [Deka 
ONT VLONDNLVMGLOR: Renters) ent tl ee eR en 337 
298. Ages of Husbands and Wives, Great Britain, 1901. Data from Yule’s “Theory 
GPO AGATA Tat Re Naetecd W eas AY A gh aR WE Pts ne Same ae eee Oe eat 338 
299; Female mcciaentMiortalityates, UsySe lL OLO=12) whe vr heen onan) 339 
303. Expectancy of Life for Adults without Tuberculosis. Data from L. I. 
LONE Oe cies Se os Sle cnt Gadus igy tisk sect Soe ORR RT Ae MEL ete en 343 


311. Duration of Employment, California, 1918. Data from Paul E. Brissendon. 351 
312. Wages and Hours of Women-workers, Virginia, 1920. Data from Monthly 


ICU OVEIRCOLCED S AORN SENN NS Re Saas Be ees Ss) 
313. Wages of Office, Sales, and Shop Workers, Ohio. Data from Industrial 

COHESION OAC Onset Reese Mens ae ote a Ge nate ae ae 353 
314. Wages of Female Office Workers, Ohio, 1919. Data from Industrial Com- 

PPRESSLOTINO ROOM TES aeeay ME EAD LORS cin erat ON ee OC LO a 354 
323. The Distribution of Incomes, U. S., 1918. From the National Bureau of 

CON CTNICMRESCATCINSARA RNC TAGN & is Sele tell ad aie Salallo ees wes Dee 361 
3 /-—-saables of the Natural Logarithms: sais 28 2 sts ve anise «alte mets 374-5 
329-31. Price of Potatoes, U. S., 1913-20. Data from Bureau of Labor Statis- 
332. Samples of Rate-of-change Chart-paper. From Mr. John Wenzel........ 392 
334. Wholesale Prices of Electrolytic Ingot Copper. From Bureau of Labor 

SR OGEAOGE ccc ign, at CRG te a Re Bees SR ae POE OE ee Pee ee 395 
335. Wages, Prices, and Money in Circulation. _ Data from Monthly Labor 

RED ICL RECS OME ATs, | SN Mats oe ane aerial Rass .8 Saige erate ow 396 
BO Om OLd rs CGold EOC IE OLOMN. ee chica tek ata «satel os (atodeloialialateeehs ahagacavoqeuavarers oh 398 
B37 2One Way to Hind the Rate-of-change scale... 2.00.2 02 00sec ee Seles os 400 
338. Annual Rates of Turnover of Bank Deposits. From Federal Reserve Bank 

Cif INU GTba DOS: us bie eect Soe ee eh AE Aan Caio cate erie Eee OL SA 403 
339-41. Farm and Factory Wages. Data from U.S. Department of Agriculture 

and New York State Department of Labor... 1... cc cce cece cess a nene 404-6 
PAOMEMENGCIC CIE VOLTA ty MALES. sein ce pore Seta erolae ia Oe olt oc lol ele tae evetame te UPN wes 409 
Baa athiage and Divorce—W.0., 108/191. stones sae steele ede tela 410 
345. Cultural Growth in the U.S.: periodicals published, patents issued, college 

students, and Ubrary volumes, [8/0-19200 0) wear oe eee 412 
346-7. Population, U. S., 1790-1910. From Irving Fisher..........-.+4-: 414-415 
348. The World’s Production of Gold, Iron, Coal, and Cotton, 1800-1919..... 417 


349. Violent-death Rates from Homicides, Suicides, Lynchings, Street-accidents, 
Railroads, and Automobiles, U. S., 1900-1920. Data from Tuskegee 


Epa CEI 26s Co IDO TECHIE nee en GO BLT aD AGUS Sno 418 
350. Retail Price of all Articles of Food Combined, U. S., 1913-22. From the 

Monthly Labor Review... 0c cv cae cent etn eee eee n ene es 420 
352. Trade-Union Membership of the World, 1910-1919. Data from Inter- 

Heol Levon QPEL «tide © «ine Radia ecleald awsome ah ae eels gy 423 


353. Consumption of Gasoline, U. S., 1911- 1930. From Pogue’s “Economics of 
IPE RUE EUS Rarmmnaace, Foyt rate bay 0 cur Resta BG O80 eds SOD BOILER LORE UC RE? ae ea 424 


XXXIV LIST OF ILLUSTRATIONS 


FIG. PAGE 
354, Vital Superiority of the Female, England and Wales, 1851-1910. Data from 
Registrar General of England and Wales.........--+++0++ss00es shat 427 
355. Output of Coal Miners, U.S., 1919. Data from Ethelbert Stewart........ 428 
356, “Duration of IWarriagesyW.91y 18/1 90Gr reas acts oe ane tee letereesteteartera: 429 
357. Sky-cloudiness, Breslau. Data from Yule’s “Theory of Statistics” ........ 430 
358. College Salaries, U. S., 1920. Data from U. S. Bureau of Education...... 431 
359. Rent Increases, Washington, D. C., 1920. Data from Monthly Labor 
Revita ee etter 432 
360. Length of Words. Data from Bowley’s “Elements of Statistics.’......... 434 
361). “Size Of Farms; {U0 S., 19205 a2 wcguc «aaron Saree te Seda een ee ete 435 
362. Size of Strikes, U. S., 1916-21. Data from Monthly Labor Review........ 436 
363. semale Mortality: Rates; S:y)LONORs. = pet yhetercieret aerate ee ees eres 439 
364 Mortality: Ratess US. 1901 1I91ON ae xref eerste eee) ara ioenaieetes wearer erat 440 
365. American Accident Table, 1919. Data from O. E. Outwater............. 441 


367. Duration of Strikes, U. S., 1916-21. Data from Monthly Labor Review..... 445 
368. Distribution of Incomes, U. S., 1919. Data from Collector of Internal 


RECRUIT OR, SR ee SR ee ee 447 
373. Samples of Probability Chart-paper. From Codex Book Company........ 458 
374, SSizetofParnis.. UaSs 1S90=191 0s eee ane cae es Oe eee 459 
375-6. College Salaries, U. S., 1920. Data from U.S. Bureau of Education.. .460-1 
377. Output of Pactoriess US: 1904-14 Wr © ate ate nice eee he eee 462 
378. Wholesale Price Changes, U. S., 1891-1913. Data from Mitchell's “Index 

Numbersofi holesalenPrsces: 2 vata hee ae a eee 463 
379. Duration of Strikes, U. S., 1916-21. Data from Monthly Labor Review.... 464 
380. Star-light: number of stars of specified magnitudes.................... 466 
381. Labor Turnover, California, 1918. Data from Paul F. Brissenden........ 467 
382=35"Outputiof Pactories, U. 9:, 1904=14 2 oie ane caren oe i 468-9 
384. Distribution of Incomes and Taxes, U. S., 1919. Data from Collector of 

TniernalRevenders = oo Gas Adc SRE Oe Oe eee 470 


385. Unemployment in the World, 1913-21. Data from Monthly Labor Review. 474 
386. Wholesale Prices in the World, 1913-21. Data from Monthly Labor Review. 475 


387-8. “The World’s'\Gommerce; 1800=1919... on ae oe er eee 484-485 
390. Cold-storage Holdings of Eggs, U. S., 1916-20. Data from Survey of Cur- 

TENE BUSINES Oak As oe dO CON Oe Ot Oe eee 488 
407. Chart for Determining ih Scales: for’ Curve-chartssctin. an Seente cee 524 
412. Chart for Solution of Quadratic and Cubic Equations. From Joseph 

Lipka’s “Graphical and Mechanical Computation.” .................-. 528 
414. Chart Showing Loads on Important Engine-frame Members. From E. A. 

Andres Re Re Se ee ee ee 530 
415. Chart Showing Proper Current Density for Copper Transmission Lines. 

From BY BQH 00d io io erie ees TE ee en 531 
431, Chart Showing the Proper Size of Type.........0..0.+.ce+crsveneaece 557 
432. Chart Showing Effects of Off-center Holes in Phonograph Records....... 558 
438. Chart for Determining Scales of Parallel Nomographs.................. 573 
439. Chart for Determining Scales of Zig-zag Nomographs.................. 574 
440. Chart Showing Bond-Yields. From Prentice-Hall, Inc.............0... 576 
441. Chart Showing Force and Velocity of Winds......................+-. 578 
442-3. Slide-rule and Magnifier. From Keuffel and Esser. SAAS nO OTS 
444. Slide-rule for Measurements of Beltings. From Carl G. Barth. i arese eter 580 
445. A Circular Slide-rule. From Keuffel and Esser... ccc ccceccuccuccuccs 581 
447-8. Slide-rule showing Costs of Book-printing..................seeceeee 584-5 
446, Slide-rule for Power-plant Calculations. From Walter N. Polakov........ 582 
452. Chart showing Fat, Protein, and Carbohydrates in Food. From Malcolm 

C. Roriy Poco: ¢ Ps Ren oe aes ate ener eee 591 


LIST OF ILLUSTRATIONS XXXV 


FIG. PAGE 


453. Settlements of Strikes, U. S., 1916-21. Data from Monthly Labor Review.. 592 
460-66, 472. Occupations of the Gainfully Employed, U. $., 1920. Data from 


Ca SHB ureauiop Lavon Statistics... ee einen . .600-05, 611 
467. Wages in Manufacturing Industries, Ohio, 1919. Data from Industrial 
COMMS SLORIONL OO: em eM Bene Hees Sek Ricoto wooo eerie. 606 
6c: Occupationsiof the Populations: Us Ss 1918s. 6.0 soso neces eencedecnce 607 
469-10 aa World's:Coal’supply (Unmined)... 02. os. s00-ce.0002e eck esosss ates 608-9 
A/mm ewish Lopulation of the World, 1920... 3....2.e0cm- 0.6.1 e tren: wy 610 
473. Wholesale Sales, 1922. From Federal Reserve Bank of New York......... 615 
474. Average Incomes of Tax-payers. Data from Collector of Internal Revenue.. 616 
475. Earnings of Corporations. Data from Collector of Internal Revenue....... 618 
476-7. Charts Showing Equations of the Quantity Theory of Money. From 
GSE UONIT aborts GB a See C EO ee 620-1 
4jcam Map showing. V.alueiot barmeland, UW. S.c).< .sscclehe ssn ose neon pn ondaes 624 
Ac OmeViap showing, Population. ofotates.. «acs sas» 4s-0ro« oases ao 627 
481. Model Showing Gas-mixtures for Gas-engines. From John B. Peddle’s 
BOORSTEUCHOIMOf GLa Phical Chanisccar: ae. cinc cr lane. oo Mae ee . 634 
482. Collapsible Model. From John B. Peddle’s “Construction of Graphical 
ONSET esceite 8 eer ho ee ale RRR RET Eo BENT ECT Ae OE i eer 636 
483. An Axonometric Model-Chart. From John B. Peddle’s “Construction of 
GLEPIICOVEHAT Seened Pete te 8 ic TPE Ce TAR ea 640 
484. Tables of Scales for Axonometric Charts. From John B. Peddle’s *‘Con- 
KPLELLOIPOPGTAPNICGUGHGIES fort kee, eer se ty Saas cin ob ee oie 641 
Zoome\Viap Showine: istribucion-of Cattle, Us So paou..ne.0 002.6 cow secu ees 646 
487,89,91. Wet and Dry Months of the Year..................... 650, 654, 657 
488. Stature of Fathers and Sons. From G. U. Yule’s “Theory of Statistics”... 653 
490. Model Showing Cost of Electric Lamps. From R. E. Scott............- 656 
492. Model Showing Efficiency of Copper-alloys. From John B. Peddle’s 
OOM TUCHLOMIOL Gia PhIGal GHATS oe see cetera oe ee Sa ne Aer 658 
493. The Normal Frequency Surface. From G. U. Yule’s “Theory of Statistics.’ 660 
{Stee Nap Showing ochoob=Lrancy, U.19-, 1920.5. ements es are ee eee 667 
495. Map Showing Density of Traffic in Chicago. From Haskell’s “How to 
JANE CCT! OIG CCH ADUGS C1 Lee eo) poke OCA an te IO Oe aOR Oc 669 
496, Floot-plan of a Small Statistical Department...:..............-6:..5--- 688 
497. The Lettering Pen. From Keuffel and Esser... ...cevceveevvevcceseees 692 


498. Optical Illusions. From the Grolier Society,...sesccvvvveveeseeneeeeees CLL 


INTRODUCTION 


-In and since the War the use and development of charts 
has been almost phenomenal—so large, indeed, that at least 
one able economist who is interested in such things thinks 
that we as a country have gone chart-mad. But this develop- 
ment has not been confined to this country, and it has a very 
solid basis in practical utility. There is little question that 
the chart represents a genuine saving in time and in mental 
effort. 

In this it does not differ from the ordinary map. Suppose 
the mariner, the shipping clerk, or the school boy had to locate 
a given point on the earth with a statement, let us say, that it 
was two thousand miles southwest from London, twelve hun- 
dred miles south of New York, eight hundred miles north of 
Rio de Janeiro, and so on. All of this information might be 
useful and even, for certain purposes, necessary. It is, so to 
speak, the statistical data of the question. But it yields no 
picture. A map or a globe gives us this mental picture almost 
ina flash. And that is precisely the use and service of a chart. 
Let us take an example: 

Within the last few months from this writing, the news- 
papers have been filled from day to day with reports of this or 
that industry making a “‘new high record.” ‘The figures give 
the idea of a prodigious boom, and, as we have so sadly learned 
to know, practically every boom is followed by a crash. So 
the wise man will shake his head at these “new high records,” 
and sagely observe that “it cannot possibly last.” 

Well, in most industries with which we are acquainted, 
such new high records are the normal and usual thing, and the 
absence of them the abnormal. In other words, practically 
every industry, just like the population of the country, has a 
fairly steady rate of growth, and so, with sharp interruptions 
that come at more or less irregular intervals, it is the normal 
and characteristic thing that they should make these new 
high records. Naturally, such high records should at least 
not be regarded in the light of sensational news. 


XXXVIl 


XXXVIII INTRODUCTION 


Let us take our old friend pig iron as aninstance. We have 
monthly records of pig iron production running back for forty 
years. In twenty-four of those forty years some month of 
those years has made a “new high record” in pig iron produc- 
tion, that is, in 60 per cent of the cases. 

Furthermore, these new peaks of production tend to run 
in sequences of four, five, and six years. So if we see an esti- 
mate that pig iron production for this year, let us say, will 
“break all records,’ we know that this is a rather foolish way 
of putting it, that it is simply the fairly normal thing and what 
we might reasonably expect in the absence of any powerfully 
disturbing causes like a world war or a profound depression in 
trade. 

Now all this information you may laboriously dig out of 
the actual figures if you like, but you can get it all in a quarter 
or maybe a tenth of the time if it is spread out in chart form. 
Like the point on the map, all these relations there stand out 
vividly and almost instantly. 

But it is not alone the economy of time and effort that is 
involved. The great thing, often, is that the chart will flash 
the thing not merely to the eye but to the mind; I mean that 
the picture gives you the idea of making the computation, 
and even that there is such a thing as a norma! rate of growth, 
as in pig iron production. Lacking the picture, we might have 
little to prompt us to make the investigation or suggest even 
a hypothesis. 

I know there are those to whom this easy method of mental 
traveling is not attractive, and even, perchance, a little irri- 
tating. Nothing else could explain, for example, why it is 
that our mathematicians should often go through long and 
laborious calculations in an endeavor to find out whether any 
close correlations exist between two sets of data, or whether 
a periodogram is going to fit a given set of figures sufficiently 
to make it the basis for a forecast, when there is a far quicker 
route. While recognizing to the full extent the value which 
these methods may have in competent hands, it is still literally 
true that thousands upon thousands of calculations of every 
kind and description have been made as to these degrees of 
correlation and all their like, involving hundreds and even 
thousands of hours of needless and useless work, when a near 
approximation in ninety per cent of the cases could generally 
have been obtained with a log chart in much less than an hour, 


INTRODUCTION - XXXIX 
The typical mathematical bent of mind seems to luxuriate in 
difficulties, long calculations, and complicated formulae. 
The simple, swift, and direct seems to be foreign to its nature. 

In our work at the Bank, we have had much reason to 
study attentively these normal rates of growth. It is quite 
astonishing to find how characteristic they are of the different 
industries, and different lines of trade, and even such things 
as the growth of bank deposits, money in circulation, and 
numerous other fluctuations of the modern economic world. 
They are so characteristic, in fact, that very often a log chart, 
with the figure for the average rate of growth in, let us say, 
the last twenty years, will suffice to identify the subject of 
the picture without- further label. 

But this idea of the persistence of growth, as a kind of a 
characteristic inertia in the different industries and trades, is 
certainly foreign to our present ideas about business or the 
thought of many economists. There are as yet few of our 
business men or industrialists, for example, who are now 
willing to believe that one can make a fairly good guess as 
to, say, the average production of pig iron, or the average 
railway traffic, or the average postal receipts for the years of 
1930-33. It is almost certain that few industries or few enter- 
prises are now planned with any long look into the future. 

There are very notable exceptions, like the American 
Telephone and Telegraph Company and others that might 
be mentioned, where the work of development is planned out 
for years ahead. For most men, even in our large industrial 
enterprises, these are pretty much matters of rule of thumb or 
of year-to-year pressure. If it were not so, we should scarcely 
have such violent ups and downs of production and trade, the 
booms and slumps that bring such demoralization to industry 
and to profits, and so much needless suffering among the wage- 
earning population. rs 

Some day we shall find a way around such stupidity, and 
it is my own belief that the most accessible avenue is through 
the grouping of the available data into interesting and well 
conceived charts. They are the most reliable and most 
stimulating instruments of education that we possess. 

So I think it has been a worthy service that Mr. Karsten 
has performed in writing such an encyclopr ac and exhaustive 
work upon the subject. The time is night zor it, and it should 
be highly useful. I do not mean to suggest by this that the 


xl INTRODUCTION 
+ 

mere making of charts is the whole story, any more than the 
possession of a fine hammer and a chisel makes a good carpen- 
ter. But it is certain that, without good tools, the best of 
artisans is badly handicapped, and I believe this is equally 
true of the business man and the director of large enterprises. 
He cannot but be going somewhat blindly if he does not have 
at his right hand, maps and charts of his whole work, extending 
years into the future, so that he may plan and anticipate in a 
truly prescient way. 

The rest of the story is that such scientific recording and 
projecting into the future makes of business and industrial 
enterprise a kind of romance in reality. Even the most 
interesting of occupations gets to be a kind of humdrum 
routine, if we have no long look ahead. Nothing stimulates 
the imagination more than a well constructed excursion into 
the future. And in business enterprises this is almost im- 
possible without the intelligent use of charts. 

But there is more. So prodigious have our industrial 
activities as a nation become, so varied and so diversified, 
that it is given to few men, even the ablest, nowadays, to 
maintain any accurate and adequate idea of current business 
trends and developments, and carry on their own work at the 
same time. So I believe that soon our successful captain of 
industry, like the captain on the great ocean liner, will have 
always at his elbow a trained navigator or business pilot, who 
will supply him with the material wherewith to study his 
course and make his plans, and who will tell him at any given 
moment just where he is at! And such a navigator will find 
his most useful tool to be a first-hand working knowledge of 
the different forms of charts which this book describes. 


Car_ SNYDER. 
New York, 1923. 


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PART 1. NON-MATHEMATICAL CHARTS 


CuaptTer | 
MAPS AND DIAGRAMS 


It is probable that the original diagrammatician lived 
many centuries ago, and it is not impossible that he was a 
cartographer. A search for him would lead us back to the 
days of “Captain Kidd” legend, when, judging by some 
records, the word “chart” invariably connoted a faded sketch 
of a lone island, dead trees, and buried treasure. It would 
lead us back to that intrepid explorer, Marco Polo, whose 
revisions of geography upset his contemporaries; back to the 
Arabs, whose excellent charts of the skies played so large a 
part in their nocturnal travels over the desert; and back to 
the Phcenicians, who doubtless kept strange maps to guide 
them about the Mediterranean shores and perhaps to warn 
them of dangerously shrewd villages where the bargaining 
was not profitable. We could not stop at the Egyptians, 
four thousand years ago, whose floor-plans of the pyramids 
have recently yielded up to us their secrets, nor at the Chinese 
whose six-thousand-year-old maps of the heavens have con- 
firmed modern astronomical calculations of star movements. 
We should be carried back to prehistoric man, at least sixty 
thousand years ago, some of whose drawings have been iden- 
tified as diagrams of familiar constellations. In short, the 
antiquity of maps is well established. 

Not only are maps the oldest form of charting, but to this 
day they are the most widely understood. And the subject- 
matter they portray is of the greatest variety. Few, even of 
those who use maps regularly, have any idea of this diversity. 
Of the United States alone, there are on the market special 
maps showing the natural resources, the density of the popu- 
lation, the location and amount of the various crops, the chief 
centers of the various industries, the lines of communication 
and transportation. Some maps show political divisions, 
others the physical contours, others the mineral subsoil, and 

I 


CHARTS AND GRAPHS 


Sr 
ce) 
hs) 


ie 
ZiT ~~ 


~ 


By Mr. C. Van de Wall. 


Pictorial Map of the United States. 


Fig. 1. 


MAPS AND DIAGRAMS 3 


still others the atmospheric conditions. Some show railroad 
distances between cities, others show automobile distances. 
It would be difficult to find any important phase of American 
life for which somewhere a map is not being published and 
marketed. 

The student of maps will note that wherever large sections 
of the earth are shown, the map seems to suffer a distortion of 
outline, so that two maps of adjacent territories will not fit 
closely together and form a single large map. He will recog- 
nize that this is due to the fact that the earth is a sphere, 
while the map is printed upon a flat surface. We are indebted 
to one Christopher Columbus, who proved that the earth is 
round, for the necessity of this distortion. The result is that 
only upon globes can outlines be truly represented. All flat 
maps being more compressed, as it were, in their centers, and 
expanded at their edges, the outlines are consequently warped. 
So, too, it follows that maps of large areas, such as the United 
States, differ considerably in shape, according as the map is 
an imaginary picture of the country from a position above its 
southern, northern, or other parts. 

In maps of the world, this distortion problem has become 


From Bartholomew's Atlas. - 


Fig. 2. Heart-shaped Map of the World. 


is CHARTS AND GRAPHS 


well-nigh insuperable, for world-maps must show us at once 
all of the earth’s surface. Imagine seeing all sides of an apple 
-at once! The most usual representation meets the difficulty 


Permission of Rand, McNally & Co. 
Fig. 3. Mercator’s Projection of the World. 


by magnifying the polar regions and spreading before us the 
sides of an imaginary cylinder. As the earth is not a cylinder 
and its poles are not as long as its equator, but are merely 


ASICS 
’ a Y 
Ravan 

WY 


/\ l ih } 


SSeS 


Permission of Rand, McNally & Co. 
Fig. 4. Hemispherical Projection of the World. 

points on its surface, the amount of distortion can be seen to” 

increase gradually from the equator and to become infinitely 


MAPS AND DIAGRAMS 5 


great at the two poles. But by increasing the longitudinal or 
_ north-and-south dimensions equally with the intersecting lat- 
itudes, local outlines over small sections of the map are reason- 


Fig. 5. Elliptical Projection of the World. 


ably preserved. This device is called ‘‘Mercator’s Projection.’”! 
Only when areas at unequal distances from the equator are 
compared, does this map become grossly deceptive. Who can 
forget his earliest impressions of Greenland being larger than 
Australia, or his amazement at the size of Canada and his 
wonder at the enormous reaches of Alaska, as gained from his 
world-map at the beginning of his school atlas? 


Ld LE BSS, 


- 
Senueeeeseey 


Adapted from maps of the Hammond Map Company. - 
Fig. 6. Homolographic Projection of the World. 


1 Invented by Gerardus Mercator, a Flemish mathematician and geographer 
(1512-1594), in 1550. 


6 CHARTS AND GRAPHS 


Such projections, of course, have no uniform scale of miles, 
for the inch that represents a thousand miles at the Desert of 
Sahara will represent but a few miles near the North Pole. 
A different form of distortion, a combined skewing and warp- 
ing, takes place in the less common maps shaped either in two 
circles or in one flattened circle or ellipse. The best preserv- 
ation of true outlines and areas, that is, a more uniform map- 
scale, is secured in a recent form of world-map called the 
“orangepeel projection,” while for less broken outlines the 
“butterfly-map”’ and the homolographic projection? may be 
found useful. These are ingenious devices to keep recogniz- 
able shapes, each gaining its advantages only at the cost of 
simplicity and continuity. 

The problem of distortion due to the earth’s curvature 
tends of course to disappear as the areas chosen for presenta- 
tion on the map become smaller and smaller. In large state 
maps it is still present, but the problem in county maps is 
rarely seen, so that several county maps can be fitted exactly 
together. Likewise a “scale of miles’ holds true throughout 
the map when the area is small. Township and city plans are 
maps of still smaller surfaces and, indeed, the category of car- 
tography? is not complete until we include floor-plans and 
diagrams of buildings and rooms, and the like. These are 
familiar in the form of architects’ blue-prints and differ from 
maps proper only in that they can be quickly prepared by 
anyone, their subject-matter being of such limited space as to 
require no professional engineering surveys. However, they 
are in principle the same as maps, in that they are likewise 
representations of space in the plane of the earth’s surface. 

These plots, plans, and diagrams of small areas are so often 
of great value that we shall here explain in detail how they 
may be made. The first step, of course, is to secure the infor- 
mation to be charted. Assuming that you wish to draw a 
floor-plan, select some convenient point of reference, such as, 
perhaps, a certain corner of the elevator-shaft, and from this 
point of reference measure the distances to the various objects 
you wish to show on the plan. Measure these distances not 


® The Encyclopedia Britannica, for example, lists some twenty-five different pro- 
jections for maps of the world, of which the most distinctive have been here described. 


* Cartography, according to the Century Dictionary, is the art or practice of 
drawing maps or charts (that is, marine maps). 


MAPS AND DIAGRAMS 7 


directly to the objects, but along lines parallel to the sides of 
the room. Thus a certain motor stands, say, fifty feet east 
and ten feet north, of the corner of the elevator shaft. Take 
a piece of paper on which the objects to be shown have been 
listed in a column, and enter these figures beside each item, in 


Plan of Small Statistical Office 


East North 
from from 
outer outer 
door door 
Chair of Statistician 8 6 
® "© First clerk -2 53 
OP endl clerk -53 5S 
* "Draftsman #13 53 
Oe Typist *12 2 


Fig. 7. Data for a Floor-plan. 


two columns. In the first column, headed East and West 
_ enter “plus 50” beside the motor (“plus” meaning “east”’ and 


Fig. 8. Samples of Cross-ruled Paper. 
The small numerals indicate the number of spaces per inch. Many other rulings 
are published. 
“minus” meaning “west” of your point of reference, the 
elevator shaft). In the second column, headed North and 


8 CHARTS AND GRAPHS 


South enter “plus 10” (“plus” in this column meaning “north” 
and “minus” meaning “south” of your point of reference). 
To draw a floor-plan or diagram, since no distortion prob- 
lems! arise in such small areas, ordinary cross-ruled or “quad- 
rille’ paper may be used. Having prepared your data,> you 
will next decide upon a “scale’’ or ratio of reduction to use in 
the drawing, that is, what value or distance on the actual floor 
shall be represented by each space or distance between lines on 
the paper. It is important to pick a scale which is neither too 
large nor too small, so that the drawing will be the right size 
on the sheet. Suppose your paper is ruled in tenths of an inch 


Fig. 9. An Unfinished Floor-plan. 


with heavy rulings every inch, and you decide to let each 
small space represent one foot on the floor, and each inch ten 
feet. At some central spot on your paper where two heavy 
lines cross, mark the letter “O” to represent your point of 
origin. ‘This point of origin on your paper corresponds to the 
point of reference on your floor. Along the heavy line through 
this “OQ” or zero-point, to the right mark the successive heavy 
cross-lines “10,” “20,” “30,” and so on to represent distances 
east of the elevator, and to the left mark them successively 
“10”, “ —20,” “ —30,” and so on to represent westward dir- 
ection. Along the vertical heavy line through the zero-point 


4 The distortion in very large buildings may amount to several inches difference 
between horizontal distances of top and bottom floors, but does not appreciably 
affect the rectilinear outlines of floors. 

® The word “‘data’’ is used throughout this book as a singular noun, unless it 
refers distinctly to more than one body of statistics. Such a usage is not sanctioned 
by the dictionaries, but is believed to be more in accordance with modern practice in 
the statistical work-rooms. 


—S 


MAPS AND DIAGRAMS 9 


at right angles to the last, mark off the inches upward succes- 
sively “10,” “20” and so on to represent northward measure- 
ments on the floor, and “ —10,” “ —20,” and so on downward 
for southward measurements on the floor. After this, it is a 
simple matter to locate and draw on the paper each item in 
the spot corresponding to its true position on the floor. The 
scale-numbers “0,” “10,” etc., can be erased and the words 
“Ten feet to the inch” or a short calibrated line, substituted. 

Here you have all the elements of chart-making. It only 
remains to observe the nomenclature. Our first step having 
been to secure two sets of measurements for each object or 
item, one for east-and-west distances and the other for north- 
and-south distances, we may call these measurements indi- 
vidually “values” and collectively ‘“‘series.” Where, as in 
this case, there are two values for each item, let us call one of 
them the ‘‘x’’ value and the other the “y’”’ value in order to 
distinguish them easily. If a third measurement or series of 
values were present, it might of course be called the “z” value. 
In the present instance we have two, and, as will be seen 
below, the east-and-west series has been taken as the ‘‘x”’ 
series and the north-and-south series the “‘y” series. This is 
only a happen-so; we might equally well have reversed them, 
but as it is, we can now write for convenience the letter ‘‘x”’ 
over our first column of figures and the letter “y” over the 
second. So much for our data. It consists of two series of 
values. ; 

Now on the chart, the two lines crossing at the zero-point 
or point of origin are called the “axes.’’ The horizontal one 
is the “‘x-axis”’ and all values of the ‘‘x”’ series are measured 
along it, the positive ones to the right and the negative ones 
to the left of the origin. Parallel lines above and below this 
axis are called ‘“‘abscissas’”’ or ‘‘abscissae’’ and the axis itself 
. . 6“ 5 = ced 
is therefore sometimes called the “‘axis of abscissas.” The 
vertical axis is called the “‘y-axis” or ‘axis of ordinates.”’ 
Along it the values of the “‘y” series are measured, positively 
upward and negatively downward from the origin. The ver- 
tical lines parallel to it are called “ordinates.” It will be 
noticed that all points on an abscissa have the same value of 
“y” and all points up or down an ordinate have of course the 
same values of “‘x’’. Taken together as a criss-cross pattern 
of lines, the abscissas (or horizontals) and the ordinates (or 


verticals) are called the “co-ordinates” of the chart. 


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MAPS AND DIAGRAMS II 


Abscissa 


Ordinate 
Ordinate 
Ordinate 


Abscissa 


Abscissa 


bscissae 


Abscissa 


Abscissa 


Ordinata 


Abscissa 


COORDINATE RULING 


Fig. 11. The Nomenclature of Co-ordinates. 


The student will observe that every point on this paper 
has two values, one along each axis, and that to identify or 
locate a point both its values must be given. He will observe 
that the axes cut the paper into four quarters (or quadrants), 
in the upper right-hand one of which (in our diagram the 
north-east quarter) both values of every point are positive, . 
while in the lower left-hand quarter (south-west) both values 
are negative, and in the two other quarters one value is posi- 
tive and the other negative. He will observe that, disregarding 
plus and minus signs, at each side of either axis, the values 
along the other axis always mirror themselves. 

Many American cities are laid out in this checker-board 
style. In New York the north-and-south roads are called 
avenues and the east-and-west roads streets. In Washington 
the former are designated by numbers and the latter by letters. 
In both cases the house-numbers began at certain axial roads 
and read away in both directions. ‘The conception is the 
same as that of the system of co-ordinates in the chart. Nor 
is it changed when we place the point of origin at a corner 
instead of in the center, that is, restrict the chart to one 


12 CHARTS AND GRAPHS 


quadrant, and thereby eliminate mirrored duplication of values 
and the need of plus and minus signs. This is commonly done 
in commercial maps, each map having a series of letters and 
numbers about its edges, the letters on two opposite sides and 
the numbers on the other two, each locating positions on one 
of the two axes. In the index or list of cities on the map, 
corresponding to our data, the proper combination of letters 
and numbers for the map is given to enable us easily to find 
any particular place. In short, the thoughtful reader will see 
that the fundamentals of charting are already familiar ideas, 
and will not allow a less familiar terminology of axes, abscissae 
and ordinates to confuse him, 


Cuarter II 


CLASSIFICATION CHARTS 


From the portrayal of space-relation between objects, we 
turn naturally to the portrayal of idea-relations, and to the 
relations of abstract ideas having no space-existence. Instead 
of location on an actual surface, we wish to show position in a 
more or less ideal scheme. We now deal, not with a geo- 
graphical, but with a logical analysis. It is not possible to 
illustrate all the uses of charts in diagrammatic logic, but the 
classification chart is sufficiently suggestive. 

In all chart-making, the material to be shown must be 
accurately compiled before it can be charted. For an under- 
standing of the classification chart, we must delve somewhat 
into the mysteries of the various methods of classification and 
indexing. The art of classifying calls into play the power of 
visualizing a “whole” together with all its “parts.” Even in 
the most exact science, it is not always easy to break up a 
whole into a complete set of the distinct, mutually exclusive 
parts which together exactly compose it.!. A child can tell us 
that the United States is a single nation (whole) composed of 
forty-eight States and a District (parts), but almost everyone 
will find difficulty in deciding the number of territories, pos- 
sessions, and spheres of influence which also compose it. 

A second problem arises when each of the parts is in turn 
considered as a whole and its own parts analyzed.2 Thus the 
State of New York is composed of 62 counties, that of Mary- 
land consists of 23 counties and one city, andthe counties are 
OE ESE SY LEED SES te a SIT Se TEN tN Natta i II 


1 The division of a whole into many parts is sometimes called polychotomy; 
dichotomy and trichotomy are cases of division into two and three parts. 

2 The decimal classification is a case of repeated subdivision in which decimal 
figures (with or without the decimal point) are used as symbols or keys to the parts. 
The Dewey decimal system for book libraries is a familiar example of this method and 
the expansion thereof by the Brussels Institute Internationale de Bibliographie, 
founded by Senator Henri La Fontaine, is the greatest achievement in classification 


the world has ever known. 
13 


14 CHARTS AND GRAPHS 


variously divided into townships, boroughs, incorporated 
places, and so on. Even a child knows that a dollar is theo- 
retically divided into ten dimes, each dime into ten cents, and 
each cent into ten mills. But no two botanists agree in the 
classification of flowers, for example, into families (wholes), 
genera (parts), and species (sub-parts), not to mention the 
elaborate hierarchies of orders, classes and divisions, and the 
multitude of sub-species, sub-sub-species and hybrids. 

The classification chart clearly presents, however, just so 
much of this marshalling and regimentation of ideas and objects 
as its author has clearly in mind. It is a method of presenting 
his scheme of things instantly and interestingly. Let us 
assume that he has settled his classification, has reduced it to 
writing, and tabulated it with indented margins or some other 
device to make it clear, and let us proceed to the technique of 
its charting. 

The simplest form of chart showing a whole and its parts 
and sub-parts is the box-chart. It is composed of squares, 
rectangles, circles, or other “‘boxes”’ arranged in serried ranks 
down its page. Across the top, a single very large box carries 
the name of the total group (whole), In a row beneath it and 


CHARTS 
NON-MATHEMATICAL 
CHARTS 


ec ene eee 
: . 


COMBINATIONS 


MAPS AND CLASSIFICATION 
> cous: 
DIAORAKS: CHARTS ROUTE CHARTS BINATION 


Combinat tons 


: Idea 
Spece Time relette 
relations relat jone saa einple cherts 


ae PROCEDURE TRE 
DIAGHANS. CHARTS CHARTS: 

Small areas lotion throug}! Motion through 
Space Time 


Fig. 12. A Simple Box-Chart. 


of 


CLASSIFICATION CHARTS 15 


tied to it by connecting lines are several smaller boxes, each 
bearing the name of one of the primary subdivisions (part or 
sub-total).3 Beneath these again is a row of still smaller boxes, 
each similarly connected to one of the boxes in the row above 
and labelled with the name of one of the secondary subdivisions. 
The process may be continued indefinitely downward, to sub- 
divisions of lower and lower rank. 

‘Sometimes there are so many subdivisions that they cannot 
all be shown side by side. In this case there are three courses 
open to us. The method most frequently employed happens 
to be the least desirable. It consists in dropping some of the 
minor boxes down to lower levels and connecting them vertic- 
ally with the boxes above them. - The method is unsatisfactory 
because it complicates the reading of the chart, changing the 
significance of a lower positioning on the page. When this 
method must be employed, it is well to distinguish the different 
ranks by various shapes, sizes or colors of boxes. 

A second method is preferable. It consists in using very 
deep and narrow boxes for the minor subdivisions which must 
be crowded together. The labels will, of course, have to be 
written downward in these boxes; they can, however, be hung 
diagonally so as to make the reading easier. A third method 
is an outgrowth of the second. In it the entire chart is thrown 
over upon its side. The main or total box now appears at 
the left of the page instead of at the top: and the process of 
subdividing is carried out to the right, each rank in a different 
column.4 This method is limited to cases of few ranks. Both 
the second and third methods are sound in principle, the sig- 
nificance of relative positioning being adhered to throughout. 

The square or rectangular type of box is the best, being the 
easiest to draw and the clearest to read. Often it is perfectly 
feasible to omit the boxes entirely, taking care to keep the 
printing in box-formation. Where two or more distinct classes 
of objects are thrown together in a single chart, such as persons 
and departments, it is a happy thought to give one shape of 
box, such as a circle for persons, to one type of object, and a 
totally different shape of box, such as a square for departments 

8 The word “‘sub-total”’ is here used in its strict sense as an inferior or subordinate 
total, a part of the grand total which can itself be viewed as a whole and split into 
parts. It is not used in the customary accounting sense of a cumulative. | oe . 

4 Tf a mathematical chart showing the value of each of the final subdivisions is 
desired, the bar-charts described in a subsequent chapter may be used in conjunction 
with a classification-chart in this form, 


et ee 


16 CHARTS AND GRAPHS 


ENCORE iN NATIONS 


GENERAL COVENANTS 
AND AGREEMENTS 

f. CONSTITUTING 

| A LEAGUE OF NATIONS 


AND CREATING 


WORLD 
EXECUTIVE COUNCIL 


creating 


WORLD COURTS 
Tribunals of Arbitration 
Councils of Conciliation 


WORLD CONGRESS 
establishing 
International Law 


Commission on 
Reconstruction of 
Devastated Areas 


Commission on 

Economic Co-operation 
of Nations within the 
League 


Commission on 
Trade Routes 


Commission on 
The Balkan States 


Commission on 
The Far East 


Commission on 
The Near East 


Commission on 
Equatorial Africa 


Other Commission 


WORLD POLICE FORCE 


Courtesy of Mr. Sidney Gulick 
Fig. 13. Chart with Boxes of Various Shapes. 


to the other type. Such differentiations should have a definite 
purpose, however, and must not be introduced merely to em- 
bellish the chart, as they then invariably complicate its reading. 

Connecting lines may be either curved, straight, or rectil- 
inear. The last are usually by far the best, especially where 
the boxes are rectangular. Straight lines, running directly 
between the boxes, give a radiating effect and are sometimes 
good when the boxes are circular. Curved lines generally fall 
into the class of pointless and undesirable embellishments, but 
are occasionally useful in complicated charts to connect boxes 
across other connecting lines. Ordinarily, when the lines 
cross, small semi-circles at the intersection on one line suffice. 


CLASSIFICATION CHARTS 17 

In drawing the boxes and connections, full continuous lines 
will naturally be used, but it sometimes happens that certain 
parts of the chart are only remotely related to the main body 
of the chart, or perhaps belong to a different period of time. 
In this class, fall contemplated future additions to the existing 
scheme. . Here the use of broken or dotted or even wavy lines 
is of value, not only for connection-lines, but also for box out- 
lines. Another means of differentiation, discussed later, is the 
use of color or shading. It is somewhat more diverting to the 


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Showing the Structure of a large Merchandizing Organization. 


} SERVICE 
COMPANIES 


; 
| 
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i LAND 
| OEVELOPMENT. 


STOCKYAROS 
COMPANIES 


CATTLE LOAN 
| COMPANIES 


MISCELLANCOUS 


| 
| 
| 


RENDERING 
COMPANIES 


COTTON OIL 
COMPANIES 


MISCELLANEQUS 


Omit OHS 
apy [OAL Thawte) 
tr Pe caeanns CO 


ipat00 CANO CF 
ORTLAND. OME, 


oo eeeesornre N 
[Puasa 
3 [Enea itn . 
|: NOS AN 


0 
ae 


Enimmeners) » 


Limo MOCO SOR] yy 
Dent Bm 


CHART 


showing 
JOINT INTERESTS OF 
THE BIG FIVE PACKERS 


{Basedon stock ownership except inthe cave of Banks and Railroads 


ear 
Coo od 


Fp ar eawcnio 
> [Sarvoa nt 60 


Tirean PG Os 
MEO it OH 


clorates} 


iY Foe Raat CO] 
fers 6Y nO ME 


which are based on dire 
= 
= 
or 


(j 


/ 


« GEES ~ 
» [ERE ASD 
era ARS 
18 | stuca yanos Co 


ise i 
ape cay 


ro ere oot 
Spat 
79 Lowes S7OUF Yd) 60 
Toman LV, WAG 
rove is0s SAK 


a Saves wa 
“ 
ewer 
TAIRA STATE BR 
rovtuana,oat_| 


RARE A Seed 
\ 


Ni Site mo oar] 4 
= pee 
\ 


(OwAL Box CO 
coucata J ** 


YSN ICE CONG) 9, 
wnt by MO 90. 

+ fiers caximo) 
IOAN CO, CAAA ND, 


TaTEN MEAT CO] gg 
FRAME ISLO,CAL | 7] 


He VBR FACING CO 
5180, 0Wn0 BY By 7h 


URION AAT 
PORTLAND | 


To 
one? 


areal 
#)1 Hold interest of steiz percent. 
e)I Hold Interest of 221026 per cenh 
*)M See cpposite page 
"IM Owned by International Products Co, 
in which Armour & Wilson {Sul ymEK seh) 
interests own 19 Fer cent 


[Ce simcet man co 
\a rey va) 


mrt 
FATUGAL ARTE CO) soy, 
Carreras} 


Reprinted from ‘“‘Summary Report of the Federal Trade Commission on the meal-packing industry, 


PUBLICATIONS: 


) 


TERMINAL RR, | 
KFACILITIES ff 


at 
‘STOCK YARDS 


PACKERS” 
MACHINERY 


SUPPLIES 


COLD STORABE 
WARE HOUSING 


COMPANIES 


\ Senay *L SLAUGHTERING 
\ 


RAILROADS 


[tasrrant ne wp CANNING COMPANY 
3 
MISCELLANEOUS 


Shee SLAUGHTERING CO. 


July 3, 1918." 


reader, a dubious advantage, however, and it cannot well be 
reproduced. 


The variations of the box-chart which are occasionally seen 


Fig. 15. An Example of Complicated Data. 


| 


B 

c 

D 

JOINT INTERESTS x 

OF THE N 

“BIG FIVE”? PACKERS ° 
(Source: ‘“‘Summary Report of the Federal o 
Trade Commission on the meat-packing dy 
industry July 3, 1918.’’) Vv 

W 

vi 


CLASSIFICATION CHARTS 19 


KEY TO THE CHART 


Bank 
Canning Company 

Land Development Company 
Packing Equipment Company 
Cattle loan company 
Miscellaneous 

Rendering company 

Cotton oil company 

Publishing house 

Railroad 

Slaughtering company 

Terminal railroads and facilities at 
stockyards 

Public service companies 
Coldstorage and warehousing com- 
panies 

Stockyards companies 


Fig. 16. Five Interlocking Classification Charts. 


are usually inspired solely by a desire for an artistic appearance 
and raise in turn some doubts about the material they present, 
that is, doubts as to its accuracy or the spirit in which the chart 
was compiled. The “‘tree-chart” is a sample of this, in which 
the trunk of the tree represents the total group, the branches 
the primary subdivisions, and the leaves, twigs, or fruit the 


20 CHARTS AND GRAPHS 


minor subdivisions. Another is the “planetary chart,” in 
which a central sun is labelled the whole group, its planets the 
primary subdivisions, and their satellites, either encircling 
them or outside of them, the minor subdivisions. Such varta- 
tions are justified only when something in the nature of the 
object shown suggests a particular fitness in the figure. The 
variation always makes for a certain amount of difficulty in 


: BROS Oy Tac 
et A 
Snakes Ligard, 4 cone 


Cheon = 
Fishes SPE Ampbion2 
= 2 \3 
axeler } “|S Lone 
g Tunes 
\-) 
d Cepos opods) 
Wa YA cero 
pS <5 < es 
SCA, - 
lnyecte S210 seat 
= meet a | eHnoderm# 
Terssiacears 
~ @efenléra 
Pesozoa V sponse, > 
Intuso St |7 Gregarines 
te nn poy 
ae = Plants, 


Te Le 


From Thompson's ‘Outline of Science," published by G. P. Putnam's Sons. 
Fig 17. Tree Chart. 


Showing the Evolution of Animal Life. 


reading the chart, it takes a great deal more time to prepare, 
and, if well done, is likely to draw more attention to its own 


arrangement than to the subject-matter it is intended to 
convey. $ 


"The student will detect in classification-charts certain elements in common with 
the diagrams described in the previous chapter. Axes are no less present because 
they are not drawn and calibrated; for in one direction the positioning signifies lower 
subdivision, while in the cross-wise direction it signifies equal and independent im- 
portance. No scale is used because the variables are not numerical measures, but 


only idealogical relations. While the analogy is not important, it is interesting to 
keep in mind. 


eae 5 me ie ey ‘ \ #= 


Cuarrer III 
ROUTE-CHARTS 


To the executive type of mind few charts make such instant 
appeal as those describing movement—that flow of goods 
through a sequence of operations which is the keystone of 
industry. Economics itself is but the study of the successive 
forms of “wealth” through the processes of production and 
distribution. Static relations, either physical, as shown in 
maps, or logical, as in classification charts, may engross the 
academic interest; indeed a correct conception of them is essen- 
tial. But when through them is woven the added element of 
time and motion, the result 1s lifted out of the field of cut-and- 
dried research and given the values of life itself. And he who 
weaves such a pattern performs, no matter in how small a 
way, a creative engineering function. The picture of such a 
process we call the route-chart. This chart throws powerful 
light on the weaknesses and advantages of a process, either 
existing or contemplated, and gives to the reader, perhaps even 
to the author, a grasp of the subject which no amount of text 
canequal. Itis a photograph capturing that highest of human 
achievements, the mental visualization of action. 

As is often the case in chart-making, preparing the data 
for this chart is no small part of the work. The data consist 
of the accurate record of the steps, changes or events which 
take place. This record may be compiled in the form of notes 
or text. In simple cases the successive steps may be listed or 

tabulated, using indented margins where the process branches 
or splits into different channels. Such data are very similar 
in form to the data for classification charts, already described. 
But where the process is complicated with detours, by-products, 
cross-connections and detailed assemblings, no list or tabula- 
tion will remain clear and the data must take the form of a 
careful statement in notes, possibly in conjunction with card- 
indices, a cross-reference system and rough working sketches. 


21 


iy} CHARTS AND GRAPHS 


It is in fact not a bad practice in extreme cases to use a large 
bulletin-board or wall, and, having the information written on 
scraps of paper, to arrange and re-arrange these scraps of 
paper with thumb-tacks thereon, until the final order is settled. 

Sometimes apparently complicated data turns out to be a 
series of the combinations and permutations of simple elements. 
Every step or event is then merely a combination of two or 
three or more items of a descriptive nature. When these de- 
scriptive component items are broken apart and listed indi- 
vidually, it will usually be found that they are few in number 
and can be grouped according to their nature into different 
series, in such a way that one item from each series is present 
in each event. Commonly, three of these component series 
are sufficient to identify all the events. A production process, 


MAKING THE CURVE CHART 


(Operator) 


(Subject) (Operation) 


1 Data, Sources Securing of 

2 Computing Instructions for 

3 Execution of 

4 Checking of 

5 Chart, Data Inspection of Chisf 
6 Field Choice of " 

7 Data Entering of Typist 
8 Checking of Clerk 
9 Scale Choice of Draftsman 
10 Curve Plotting of , 

11 Checking of Clerk 
12 Scale Entering of Typist 
13 Chart Inspection of Chief 
14 Title Choice of us 
15 Entering of Typist 
16 Chart OeukKe Chief 


Fig. 18. A Tabulation of Simple Route-chart Data. 


for example, may be made up of operator, object, and opera- 
tion (“who,” “what” and “how’’). Time (“when”) may also 
be actually recorded. A distribution process may be made up 
of combinations of place, person and proportion (“where,” 
“by whom” and “how much”). The number and nature of 
these component series will vary widely in different processes, 


ROUTE-CHARTS 23 


but the above are fair samples. Needless to say, where the 
data can be analysed in this way, it will simplify the work of 
compilation and assure the completeness of the data to list the 
events, together with their component details, in parallel 
columns, a column for each series or type of detail. 

A still more condensed type of work-sheet can be prepared 
for complicated data, in which only two types or series of 
descriptive items are present. This consists of a diagram in 
which each series is listed fully and once for all, along an axis 


MAKING THE CURVE CHART 


Statis- | Computing| Drafts- | Typist 
tician Clerks man 


Instructions € 


Computing 


Checking 


Inspection 


Choice of Field 
Entry of data 


Chart 


Checking 


Choice of scale 


Plotting 


Checking 


Entry of scale 


Inspection 


Title 


Entry of title 


O.K- 


Sh 


Vine 19. A Condensed Work-sheet. 


or edge of the paper. Thus on paper with columnar rulings, 
list the descriptive items of one type or series (e.g. materials) 
down the left-hand edge of the paper, and those of the other 
type or series (e.g. departments) across the tops of the columns. 
Then along the line of each first-series item mark with a cross 
or circle the columns of those second-series items with which 
it is combined to make an event or step (e.g. operation). It 
makes little difference which series is put along either edge. 

1 The student will be reminded of the keys and data-sheets used for maps and 


diagrams, in which the two sets of values, or measurements along the axes, were listed 
in actu form, and will compare the descriptive detail series to the numerical value 
) 


series. 


24 CHARTS AND GRAPHS 


If one series is longer than the other, it should be at the side, 
but if both are of the same length, have the more important 
series—the series by which events are to be grouped—listed at 
the side edge of the paper. If this be reversed and the more 
important series placed across the top, the crosses or circles 
would of course be entered up and down the columns. instead 
of across them. The sequence of events can be shown by 
small numerals in the circles, or by connecting lines with 
arrowheads, or best of all, by both. These connecting lines 


MAKING THE CURVE CHART 


Statis- | Computing| Drafts- Typist 
tician Clerks man 


fee hey 
=3— 
eg 

Sana 


Fig. 20. A Very Condensed Work-sheet. 


showing sequence would run horizontally if the side-edge 
items are more important, vertically if the top ones are more 
important. In practise either arrangement is satisfactory and 
it is usual to leave the longer series listed down the page, 
because more items can be written down a page than across it, 
even when the column-headings are entered on edge. 

The phenomenon of motion involves two inseparable ele- 
ments, space and time, in either of which the motion may be 


6 


ROUTE-CH ARTS 25 


measured. It is ordinarily simpler to prepare fitst an analysis 
of movement through space. This will include changes of 
location, condition or operations. It is, in fact, simply a 
measuring of events by arbitrary differences in their nature, 
instead of a measurement by differences in point of time. 
When it has been completed, it may also be desirable to 
measure the movements chronologically and to co-ordinate 
them upon the chart so as to show graphically the. motion 
through time as well as space. - Assuming that the data for 
the chart have been decided upon, we shall proceed to its 
graphic presentation, beginning with the simpler route-chart, 
showing only change of place or condition. For convenience, 
this may be called the “procedure-chart.”’ 

The simplest “procedure-chart”’ is a straight line or row of 
“boxes”? with the steps or events inscribed and with arrows 


THE MAIN STAGES IN CURVE CHART MAKING 


Comput ing 


Fig. 21. The Simplest Procedure-chart. 


along the connection-lines indicating the direction of move- 
ment. It is important that the arrangement of the steps be 
in a uniform direction across the paper. They can be arranged 
horizontally from left to right, or vertically from top to bottom 
(or even, in special cases from bottom to top). The same 
considerations will determine this direction as were noticed in 
the arrangement of the classification-chart; namely, the letter- 
ing of the boxes generally gives them greater breadth than 
height, and if there are but a few boxes, they can be placed 
side by side. However, if there are many, they can be packed 
closer one above the other. 

The procedure-chart is in many respects similar to the 
classification-chart, its main differences being that it need not 
branch or split up at each new step, and that the connecting 
lines between boxes indicate a path-way or line of motion. 
For more complicated data in which the processes branch out 
and split up, the similarity between the two charts will be very 
ereat. The use of different styles or shapes of boxes now 
becomes more advantageous, as the various steps may be 
totally dissimilar, and by adopting certain shapes for each 


; 


26 CHARTS AND GRAPHS 


1790 1800 1810 1820 1850 1840 1850 1860 1870 1880 1890 1900 1910 1920 


(RI. -X{R.I. —“XR.I. -<XR.I.) (R.1.)—R.1. —~R.I. —<R-1. —R.T.) 

Conn \ AMass —(Maes Mass (Mass > Mass Mass Mass Masa Mass (Masa 
[Nesey \Conn Conn (Gonn)—(Conn)—Conn (Conn ) Neds {NTs PXN-Ts XN. Ts PXN.T-) 
(wa. iid) KX Kid XNT. YY XGonn)—{Cons {Conn )—~{Conn Conn >—{Conn > 
[Del. (Del. )—(Del. Del. ) XN.J-) [as-y N.Y. N.Y. N.Y. N.Y. N.Y. 
(zy Wa. wa. Ma. Md. Penn)>—(Penn )—< Penn — Penn —Penn) 


N.H. NH. Vt. n.¥. Y \De1. —<Del. Penn —<Penn»—PennY \ud. Mas Md. Md. (ud. _) 


Vt. KEE vt. Onio {Ohio —< Ohio Onio Ohio )—Onio OhLo 


a] ja 


BLES EY FO Ps POUR Co ten ie Oo 


(Rank of the States at Census Years, 1790-1920) 


Ve 


Wash Col. Wash , . S.D. Col. Col. 
[Nev. \ Vivaah Y New. -D. AND. Y XS.D. )—¥S-D.> 
ev Ne | it E 


[>- EET EU: | fox1a } utah Utah —Utab 


[Ariz )—~{ariz yen.) idan )—~idend—(idend 
Mont wont Wi (iaans XNout feu, Mont) 
SSS 
S.D. )-S.D.9/\Yariz Ariz —(ariz>—ariz > 


Fig. 22. A Simple Procedure-chart with Many Items, 


type of step (e.g. operation) or for each type of descriptive 
detail (e.g. departments, persons, objects, functions, etc.), 
these distinctions between steps or events are clearly brought 


ROUTE-CH ARTS 2 
Je 


PerenoNn fs 


ET eG 


PeARd OACRS PE: 

ey ee eee P89.) Saves a mepaens P04 P1034 P-L PKS Kt-iB3 
eae pptinot advantageous Pianchetto Engliah Queen's King's Pianchetto VYan't fruges Zuckertert 

\Pianchotte Ene u ° ; © 
Opening Opening ae Eis /a0 @ Opening Opening Opening 

See p= fk Game Transp, of Transpositions of 

\ ’ \ Lane King’ afeawn Game 
eS 


(Queen's side) 


| 


Kt-933 P-033 P-25 P-QkKtS P-cB4 P-G4 P-E3 2-utS P-KBS P-EB4 
Irresulsr and not Pisnohetto Sicillisn Center ny a 
French Pisnchetito Irreqular, not 
edrantsgeous games Defence Defence ESAS , Defence Defence edventagoous 
Pb 
‘The King's}Pawn Defence 
d he 
eae A i 
ae ay a CQ 
King's 
Irregular Gensee F Bishop 
f Opening PERS 
King'a 
Kt-[|KB3 Gambit 


King's Knight|Opening 


b 
ay 
i j 
i : P-04 P=Qs Kt-KBS P-KBS 
Queen's Pawn Philidor's Petroff's Greece 


a Counter Gambit Defence tounter-attackCount er-Gambit 
t-| 
Queen's EnightPefence 


" P-EBS B-B4 B-QS Q-BH = QK2 


Irregular, not advantageous 
gacnce. 


t 
7) 9) ‘ 
, \ 
a) 4 
P-0B3 Bee, M 
Staunton's Ruy Lopes 
parce seed Enight's Game ® 
2s » 
eS. u 
——_ ‘4 $ 
vs 
B-| B4 - . 
ee Bishop's|Game = ’ 
ae = , 
- 
, \ U 
‘ 
‘ + iu 
’ rs. 4 
‘ Kt-05 B-K2 Kt-B3 P-B4 , 
\ Transposition Hungarian Two Enirhts' Tronspos ition 4 
mor’ Defence B-|34 Defence se oa 


Giuco|Piano 


GN RR i 2 


Otuse re 55S e 
oo Stace Jerome J 
Lo p-faxts 9-0 
ay Evana’|Gambit Gambit uae 
+) +) 
yas 7 B-Kt3 
Evans F Press 
declined BxfktP dectinea 


CHESS OPENINGS a eo 


We EVANS GAMBIT 
AND ITS 3-| rs 
IMMEDIATE ALTERNATIVES 


4 it 

BY 2 = 
K. G. KARSTEN ii at, 
reply reply 


Fig. 23. A Pictorial Procedure-chart. 
In which the objects or materials of the various steps are realistically pictured. 


out on the chart. Indeed, it is sometimes possible to add 
greatly to the value of the chart by picturing the various 


28 CHARTS AND GRAPHS 


VALUE OF MATHEMAT- 
ICAL THINKING 
The Exact Measurement 
of Facts 


PRODUCTION 
The Result of Universal and Cooperative Labor 


MOMENTUM TIME RATE PROGRESS TIME RATE 
The Sum-Total of Past 
Achievements eran the 
Heritage of Civilization. 


HUMAN WORK 
Time-rate the basic Characteristic 


PAST 
Shown on Machine 
Record Charts 


PRESENT 


Shown on Man-Record 
Charts 


FUTURE 


Shown on Progress 
and Layout Charts 


MEASUREMENT OF HUMAN WORK 
Only Correctly Expressed in Units of Time 


Showing the subjects treated in an article by Walter N. Polakov, in Engineering Management, 


by permission. 
Fig. 24. A Graphic Outline of Thought. 


stages realistically. At other times the chart is so simple that 
the boxes can be omitted entirely. 

A uniform direction of movement across the chart is im- 
portant, because it automatically suggests to the reader the 
sequence of events. It would better be described as a uniform 
drift; motion at right angles to this drift, necessary at branch- 
ings of the process, being immaterial. There are occasions 
when, on account of the data, it is necessary to draw a line 
backward, as is the case when seconds or by-products return 
to an earlier stage for re-treatment, but these are legitimate 
representations, suggesting actual backward steps of the 
process. At other times it is necessary to choose between 
backward directions of lines and repetitions of boxes; as a 
rule the latter is the lesser of the two evils, but if the former is 
decided on, the backward motion should be strongly indicated 
by arrows, and the connecting-lines should leave boxes and 


RKOUTE-CH ARTS 29 


enter boxes at the points they would naturally leave and enter 
if the boxes were in proper sequence. 

Embellishments, artistic and otherwise, are often met with. 
Impartial study will usually show that nothing has been gained 
by them, and that the message of the chart would have been 


DIAN REFINING COMPANY 
. SALES DEPARTMENT |—. 


€ 


consumers § 


Permission of Mr. Richard Webster. 
Fig. 25. A Popular Presentation. 


The original of this chart, prepared in colors, is provided with a key explaining 
the various channels through which influence is brought to bear upon consumers 
by the sales department. 


more strongly conveyed without them. The occasional ex- 
ceptions to this rule are special cases of data in which the 
subject-matter itself suggests the modified form as particularly 
appropriate. In the ‘“‘boiler-chart’’ the source of supply is 
shown as a large tank or boiler, and the goods are shown as 
flowing through pipe-lines to smaller tanks, cylinders, engines 
and outlets, each representing a particular type of comparable 
stage in the process. Here the reader’s imagination is fired 
by the implied simile of a familiar mechanical process, and if 


30 CHARTS AND GRAPHS 


the simile be a good one, he is likely to examine it closely and 
so visualize the process clearly. Sometimes a row of tanks or 


Requisition on US. by CG.S0.5. 


‘Purchases in France 


a= by CPA. for CG. SOS; 


8.0/8i'Unifs | OTT Eze =a 
30 Day Level 


<o-<-¥--==--~--- BB --- 44 -~--- ----------- 


|” Distributes Supplies 
/into Depots of S.0.S. 


/; ‘w= Ceneral 
} i 7 
G : . 
fs ‘ 


=== DEPOTS =— 


90 Days Supplies in France 


Y HEN 
S.0/S|Units 


eve 


RR oR eR Ze D> hPa 
SA TRIRANNT Hi FW OWEN A 
2N°ARMY 3RPARMY DETACHED UNITS 
A Freumatic Buffer-Storage at Regulating stations sufficient only to overcome 
unavoidable irregularity of shipment tram Depots & /o insure uniiorm How at 
Ralheads Permits no aver -accumulation. 

B Aai/heads. Points at which Supplies are delivered to organizations. 


Permission of Mr. Malcolm C. Rorty, 


Fig. 26. An Excellent Pictorial Route-Chart.. 
This shows the flow of supplies in the American Expeditionary Force. 


vats will represent successive cost-burdens well, the overflow 
(prot or balance) from each flowing into and feeding the 


ROUTE-CHARTS 


THE ROUND FLOW OF MONEY INCOME AND EXPENDITURE 


Income from Capital-Interest 


Bia 


Income from Personal Service. 68% 


—_ 


Dividends and Profits, 242% 


Income from 


jatural Resources, 8% 


=== INDIVIDUAL INCOMES 


v cs pa e c 

¢ s|| 8 & ge 
wn 

(eed ee ietaa oe io Bl feo 2 

el fc € ©: ie Bot art 

c Oi @o of e 2 

Bea Sere be 36 

Oo In 

ov i) i} i) 

Gy a o a 


, INCLUDING == 
ATION AND BANKING: 


G = 
3S} |3 &| 13 
eal |S ol |e 
HK 3) fs 2] |S 

S e Cc 

so] le 5 

Se] le & 

£0 

ce} | Sac 


PRODUCTION OF 
RAW MATERIAL 


for Taxes, 


Natural Resources’ 
Payments for 


t 


for Interest} 
tc, 
Payment: 


Payments| 
for Natural 
Resources, 


Permission of Mr. Malcolm C, Rorty. 


next. 


Fig. 27. The Analogy of Vats, Tanks or Reservoirs. 


There is no limit to the possible variety of such repre- 


sentations, but in the main the chart-maker will find the widest 
play of his imagination called for in the making of legitimate 


32 CHARTS AND GRAPHS 


procedure-charts, and will do well to avoid the excessive and 
wasteful effort needed for the artistry of more sensational 
products. 


‘LOADING VB RIFLE GRENADES 


PROCESS CHART 


BLACK FUSE ROSIN AND TROJAN 
/\, FUSE CONTAINERS BAGEL /Nerenace JN L\ Rosi N (\ Troan 
\ BODIES 
A WI CASES ARI 
Gi) CASES ARE OPENED jeg ela CD Geened 
FE / VARNISH FOR 
FUSESARETAKEN TO <8) TO FUSE 2 eatiglas W) inreRion OF 
FUSE LOADING ROOM LOADING ROOM EATS ORENDESIS 
FUSE POWDER 7 MIXEDHERE 
@ ee FUSES : ©) EWASHED WITH GASOLINE 
@ BELS AND STENCILS INTERIOR OF GRENADE IS COATED 
E\, PRIMERS @) TOINSPECTORS TABLE (1) ARE REMOVED BY 63) with VARNISH 
, SANDING MACHINE y 
2) TOPRIMER INSERTERS POWDER CHARGE /S GRENADES ARE PLACED 0) SR INTICSEEEMT 
Bs) inspecreD ©) Drie NS WACHINE TOREMOYE @) ED TORENOVE ANY 
o>) FUSE IS REAMED KRECTIONOP REIGN 
A Salas ® snocemreo OS) INTERIOR VARNISH SUBSTANCE 
> ISPECTION OF 
@D_TOSHELLACING TAB REAMING AND CRIMPING 6) TOTROJAN LOADINS ROOMS 
Ge) PRIMERSARE INSERTED = 
/\, Brass 6) INFUSES Gey SETADES ARE LOADED TopROPER 
poe Gp SSOAUSES ARE PLACED CEERI RE CII AG. 
GD TODISCING TABLE INEACHTEAY, &) TOINSPECTION TABLE 
€5) NES aD MES ea 2 
/A, OFANGE ECL SCT AULT 7] GRENADES ARE INSPECTED FOR PLLER PLvES 
SHELLAC Ga) VeNTCLosMNe Disc 67) DENSITY OF TROJAN POWDER /\ 
) IOSHELLACING TABLE alse (Se By TOEXTERIOR CLEANING TABLE 
. Gy VENTCLOSING DISCS © @ TOFILLER PLUG 
O72 WATERPROOFED WITH INSERTERS 
SHELLAC G8) ANYLOOSE TROJAN 1S REMOVED: 
Gi) TY INSPECTION FROM EXTERIOR SUBEAEES OF GRENADES c 
OF CONPLETEFUS 
©) ALLER PLUGS ARE INSERTED AND SCREWED UP STRIKERS 
FUSES AND STRIKERS ARE ASSEMBLED 
zy TOFUSE 
INTO GRENADE BODIES. INSERTERS 
a 


L\ DETONATOR 
PLUGS. 


LA COMPONENTS 70 PACKING 


ROOM BULLET TUBE 1S 


COATED WITH VATU-. 
DRI 


aS QUALITY INSPECTION @ TOEND OFASSEMB- 
LING TABLE 

/\ STRIPING 
ENAMEL 


@ TO PACKING ROOM 


0 euamriry inseecrion 


GRENADES ARE 
@) OPERATIONS ONFUSES STRIPED WIT 


/\ SIGNODE STRAPS 
AND SEALS 


® TOPACKING ROOM 


3 QUALITY AND QUANTITY 
K INSPECTION 


OPERATIONS ONGRENADES BODIES 


2) 24 GRENADES ARE PLACED 
INCASES AND COVER 
PLACED ONG 


OPERATIONS ON CASES 


CASES ARE SEALED 


WITH SIGNODE STRAP: 
OPERATIONS ON BLACK POWDER AND SEALS 


C3) CASES ARE STENCILED 
TOINDICATE CONTENTS 


OPERATIONS ON TROJAN POWDER 


@ TOFINISHED GRENADE MAGAZINE 
OPERATIONS ONINTERIOR VARNISH 


MAGAZINE FOR FINISHED 
GRENADES 
FINISHED GRENADES 


7 0@e0E8060 


CARRYING BETWEEN OPERATIONS 


From ‘‘Process Charts and Theiy Place in Management,” by Frank B. Gilbreth and L. M. Gilbreth 
“Mechanical Engineering,’ Jan. 1922. 


Fig. 28. A Simplified Gilbreth Process-Chart. 
Showing operations by conventional symbols and materials by pictorial drawings. 


The second type of route-chart differs from the first or 
procedure-chart in that time is an important element in the 
data and a feature of the chart. It may be called the ‘‘time- 
chart.” It is easily made by arranging a time-scale along the 


ROUTE-CHARTS 33 


axis or direction of movement of the procedure-chart, and 
adjusting the various boxes or entries of events so that their 
positions coincide with the ordinates or abscissae of their par- 
ticular points of time. The scale of time should be marked 
along the edge of the chart (in large charts along both edges) 


STEPS IN MAKINS THE CURVE CHART 


Keatertals 


Data-Sources Pind© 


Deta-Conpur a : 


Chart field 


ee 
| 
Fig. 29. A Simple Forni: 


In practice this chart would bear dates for the days or weeks or other time intervals 
at the top of each column across the page. 


ant: 


and straight lines in a faint color should be ruled across the 
chart from the main divisions on this, scale. A glance across 
the chart will then show, by means of these faint lines, how 
many time-intervals elapse between steps or events and a 
study of the scale will give a more exact estimate when desired. 
In time-charts, needless to say, the uniform direction of move- 
ment or drift is essential. 

The time-chart may be reduced to a form similar to that 
of the work-sheet already described for procedure-charts. 
With the time ruled off on one axis of the diagram, the items 
to be followed by the chart are listed on the perpendicular 
axis and the action of each item indicated by crosses, checks, 
or solid shadings along the line of this item under or opposite 
to the right moment of time. By various kinds or colors of 
shadings, or by words alongside the shadings, the nature of 
the event happening to the item can be indicated. In prin- 
ciple, it is better to place time on the horizontal axis, so that a 
standard time-scale can be used and long charts folded in 


a CHARTS AND GRAPHS 


Curves -- 1/20 to 2/5/17 
CUARVE-CHART PROGRESS RECORD 


9) 
fs oo 


a/ee 


, 


Shipments 


Werehous ing 


Diétrict 9 Sales |SM-3 


Fig. 30. A Time Record, But Not a Time-Chart. 
This is inserted for comparison with the previous illustration, as the dates form 


the body of this table instead of the column headings, and the operations form 

the column headings instead of the body. 
sideways to reduce them to the same size records or files. If 
very finely ruled paper be used, a large amount of detail can 
be crowded into this sheet and if a number of similar processes 
are to be compared, a standard arrangement of the items on 
it will make quick comparison easy. This simplified form of 
time-chart has little to recommend it from a graphic point of 
view, but it will be found extremely convenient and sometimes 
indispensable as a record, and is always useful as a work-sheet 
in preparing a more graphic time-chart. 

An example of this simplified time-chart is the Gantt 
chart method. All charts in the Gantt system employ uni- 
form vertical rulings, marking off “time’’ on the horizontal 
axis. [he particular markings are always adjusted to the 
individual business, so that the spaces between vertical rulings 
may indicate hours, days, weeks, or months as is desired. And 
the columns between these vertical lines, after their adjustment 
to the periodicity of the particular business, resemble columnar 
accounting sheets. At the left-hand edge of the paper, in a 
very wide preliminary column, the machines, departments, 


And along 


roper “time” (that 


quipment are listed. 


items, under the p 


ROUTE-CHARTS 


ls, or other form of e 
the lines of each of these 


materia 


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36 CHARTS AND GRAPHS 


of the operation and ending at the time of its finishing. This 
line takes the place of the crosses or circles above described 
and has the advantage of showing by its length the length of 
time required for the operation. The chart is used not only 
for laying out work in advance in the planning department 
and in the individual departments, but is also used to record 
actual performance when the records of the same are being 
kept. Wide lines are used in the place of narrow ones to 
record actual performance. 


CHART FOR 8 HOUR TURNS WITH 24 HOURS OFF AFTER 6 DAYS, /N CONTINUOUS 
OPERATION INDUSTRIES 


VP SECOND WEEK THIRD WEEK 
P60) 
£6 
No 


* Mon-| Tvés-\Weones| Trvrs| Far- Sun | Mon-Tues: 
oar DAY | oar @AY | oAY oar OAY | ORY 
| TURNS |\TURNS |TURNS \TURMS\TURNS; TURNSITURIVS TURNS 
7] (j2)3|7J2[3|7J2[3 1/2] 3] [2[3] -[2]3] 72/3] ]2[3| zis 
i bd | bd Tf bf | bet bet | bt | bet | 
2 B Yt bd | bd 


bd 


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s 


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es 
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7 
= 
S 
| 44 | 
| 6 | 
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es 
| A | 
Sai 


Permission of the Bureau of Labor Statistics. 


Fig. 32. A Simple Time-Chart. 


Another notable example of this type of chart is the Gilbreth 
“process-chart,” used by Mr. Gilbreth to analyse methods of 
work in his well-known micro-motion studies. On paper ruled 
in tenths of inches, each fine line of which represents one- 
thousandth of a minute in time, the length of time to perform 
each operation or element of an operation is recorded by a 
heavy line, beginning at the commencement of the operation 
and showing by its length the number of thousandths of a 
minute required to complete the operation or element of an 
operation. Mr. Gilbreth ordinarily measures time vertically 
down the page, a detail in which the present writer believes 
him to be ill-advised. Across the page his vertical rulings 
mark off the different parts of the human body whose motions 
he is studying. There are about a hundred of these parts 
listed at the top of the page, each over one narrow column or 
space between vertical lines. Obviously the lines indicating 


ROUTE-CHARTS 37 


operations, extend vertically down the page under the parts 
of the body active in the operation. Different colors of the lines 
indicate different elements of the operation and different widths 
of the lines indicate degrees of activity engaged in the operation. 

Where the time runs in natural recurrent cycles, such as 
days, weeks, months, or years, and items re-appear at identical 
points in each cycle, a circular form of time-chart is often 


Ay 
¢! 
> it 


FRIDAY 
9 
O 
SON LLL Boe 


Fig. 33. A Weekly Clock-chart. 
The weekly cycle of sales in a department store. 


desirable. It has the advantage of being endless without 
actually showing more than a single cycle and naturally 
suggests to the reader the recurrent nature of the process. 
In such a chart the time-scale runs around the edge of the outer 
circle and the time-interval lines, equivalent to ordinates, 
appear as radu from the center of the chart. ‘The items or 
events are inserted in boxes in their proper positions along 
concentric circles, those near the center having of course less 
room on the chart. Care should be taken to place the larger 
items, or events, requiring more descriptive labelling, toward 
the outside, if possible, in order that they may not be too 
crowded. Suchcircular time-charts are called “clock-charts” but 
they are not necessarily marked off like a clock; in one complete 
revolution they will show twenty-four hours for the day, or 


a \ 
\\\ 


\\) 
B 


4 
(if 
i 


Fig. 34. An Annual Clock-chart. 
The annual cycle of sales in a department store. 


seven days for the week, thirty-one days forthe month and twelve 
months for the year, according to the time-cycle chosen. 

Route-charts, like classification-charts, offer great freedom 
to the ingenuity of the author. No set rules can be laid down 
for their construction, though the general principles above 
outlined will be found always safe and helpful. If the author 
of the chart desires to modify it, he will break no iron-bound 
canons, though he will probably have to do a great deal of 
experimental work before he has a satisfactory product. The 
one really final criterion by which his product will be judged 
will be, as in all charts, how clearly, forcibly, and truly does 
his chart tell his story.. If he can pass this test better with a 
novel form of chart than with the typical and sound forms 
which have been described, he will have really invented a new 
statistical instrument and his product will be a contribution 
to the science, but the man with limited time will be well 
advised to follow and remain within the fundamental principles 
here outlined. 


CuarTer IV 
COMPOSITE CHARTS 


More fascinating than any one of the fundamental chart- 
types already described are the results developed by combina- 
tions of two or more of these types simultaneously. The simple 
types are three in number, adapted to showing between 
objects a space-relation (maps and diagrams), a topical rela- 

tion (classification-charts) or a relation in motion (route- 
charts). Any two of these relations may be shown simul- 
taneously by combining the principles of their chart-forms. 
It is only necessary to construct first one type of chart and 
then with this as a basis or ground-work, superimpose upon 
it the construction of another type, in such a way that while 
each retains its own significance, the two harmonize in details. 

Very often the two will be so closely interwoven that they 
seem to be inseparable and indistinguishable, but the student 
will always find that under close analysis they readily break 
down into two or more separate and distinct charts belonging 
to the essential types which have been described. He will 
also find that this process of breaking down a composite chart 
invariably clarifies his understanding of its subject-matter, and 
vice versa, that the more obviously the component charts are 
distinguished in the composite product, the more clearly its 
subject-matter will be understood by its readers. If the chart 
is composite, it is important that the maker should recognize 
its nature, and it is important that the chart itself should show 
on its face that it is composite. 

If you will take a map of the country through which you 
have travelled and with a heavy black pencil draw a line along 
the routes you have passed over, with circles or boxes about 
the names of places where you have stopped, you will have a 
simple form of a superimposition. Had you marked upon 
tracing paper over the map, instead of marking directly upon 
the map, you would be able to lift your second chart bodily 


39 


40 CHARTS AND GRAPHS 


Courtesy of Rand McNally & Co. 
Fig. 35. Route Map. 


off of your base chart, and would see that you really super- 
imposed a route-chart, describing motion, upon a map or chart 
of space-relations. ‘The result is a composite chart illustrating 
motion through space. 

Maps and diagrams are often used as a basis for charts 
showing motion. In fact there has recently been developed 
an elaborate technique of what are known as “‘pin-maps.’ 
These are particularly in vogue among sales-managers, who 
have to route a number of salesmen about the country and 
wish them to cover the most ground in the least time and with 
the lowest possible travelling expense. Maps for this purpose 
are mounted, and the markings upon them are made in the 
form of conspicuous colored tacks and other devices driven 
into or fastened onto the surface of the maps. In this form, 
the. same map may serve for many temporary superimpositions 
and the latter can be readily altered at will, without the labor 
of complete re-drawing, merely by removing or shifting the 
adhesive markings. The labor-saving value of pin-maps is 
so great for all kinds of continual routing work that they are 
marketed i in excellent form by various commercial firms, includ- 
ing nearly all map-making companies. 


LL 


See 


COMPOSITE CHARTS at 


Courtesy of Rand McNally & Co. 
C Fig. 36. Pin Map, 


The mounting of maps or diagrams to accommodate pins 
or map-tacks should be closely examined. Ordinarily the 
maps are mounted directly on wood, either to be framed and 
hung on the wall, or they are already fitted into flat drawers 
of special cabinets holding a large number of such drawers in 
horizontal positions. The wood-mounting is, however, a poor 
investment. Pins cannot easily be forced far enough into the 
wood to be secure, nor easily removed if driven deep, and 
sooner or later, as the wood shrinks under the punctured 
paper, individual pins will drop out, and cannot be replaced 
without complete rechecking of all data, the map meanwhile 
becoming inaccurate and unreliable. 

The ideal mount for a map, and in the long run the cheapest, 
is a construction of cork and corrugated paper-board. The 
map should be mounted directly upon a piece of cork linoleum, 
with a non-wrinkling adhesive called rubber cement rather 
than with paste or glue. The cork should then be backed 
up with two or three layers of corrugated wrapping board or 
paper, laid in alternate directions to prevent bending. In a 


42 CHARTS AND GRAPHS 


mount of this sort, the pins can be easily pushed into the map 
to their heads; the cork grips them and prevents their falling 
out, and the corrugated paper keeps their points from sticking 
out underneath. Maps can be mounted in this way at home 
or in the office, and in some instances can be procured directly 
from the manufacturers of the maps. 

Map tacks and other marking devices to attach to the 
mounted map can be obtained in great variety, suitable for 
showing a number of distinct markings and meanings at the 
same time. The tacks are small steel pins with large round 
or flat heads of cloth, celluloid, or, best of all, glass, conspicu- 
ously marked, in different colors and sizes. When they are 
inserted in the map to indicate, for example, towns on a sales- 
man’s route, the various colors can be used to indicate different 
salesmen, the various sizes can show the length of the sales- 
man’s visit, and the various markings on the tacks can tell 
the extent of the company’s business there. Similarly, colored 
string can be stretched between tacks to show the sequence 
in which they are visited, while small celluloid rings of the 
same color can be slipped over the tacks to show present loca- 
tion or progress along the route. Ring-pins into which cards 
can be fastened, are made in various shapes to hold cards at 
different angles to the map, flat-headed pins on which labels 
can be pasted, rough-ground celluloided pins on which pencil- 
markings can be made or erased, and a wide variety of other 
appliances are furnished for ingenious uses with mounted maps. 

It is, however, no longer necessary to have mountings and 
attached devices to make temporary and easily altered mark- 
ings upon maps. Such arrangements take up too much space 
and require special filing or housing equipment if used exten- 
sively. Instead, it may be desired to use flat maps in book 
or sheet form, which can be more easily carried about. For 
this purpose a celluloid-coated map is made, the surface of 
which is protected by a thin adhesive layer of pliable trans- 
parent celluloid.!. On this surface pen, crayon, or ink marks: 
can be made without difficulty and removed without injury to 
the map, while gummed paper stars, dots, and other signals 
in various colors can be attached and removed likewise. 


1 The celluloided map is really little more than a map which has been surfaced 
with a thin layer of liquid shellac or varnish. The liquid can be secured from dealers 
in artists materials and can easily be applied to any chart or map with a fine blow- 
spray; 1t forms a protection against soiling, as the surface can always be cleaned 
without removing the marks under the coating. 


COMPOSITE CHARTS 43 


The benefits of mounted maps suggest that floor-plans, 
and other diagrams could be likewise profitably mounted and 
used for pins and strings, but as a rule this is not yet a general 
practice. The pathway of goods, papers, or functions about a 
plant is generally marked upon photostats or blue-prints or 
other reproductions of one original floor-map, in various 
colored inks. The lines upon such diagrams should have 
frequent arrow-heads to indicate the direction of movement, 
as this direction is no longer shown by position on page as in 
the simple route-chart. Different colors or kinds of lines can 
be used to differentiate the pathways of various articles, but 
when a large number of such articles are to be individually 
followed, it is better to use a number of copies of the original 
base-map, one for each article or group of articles, so that the 
lines will not be too confusing. 

An elaboration of this method has been described to the 
writer by Dr. C. W. Gerstenberg, who tells of ‘‘a factory where 
a bird’s-eye view of each machine has been drawn to scale and 
fastened with brass paper-fasteners to a piece of cardboard cut 
to scale to represent the amount of floor-space needed for each 
machine. The color of the card-board indicates the nature 
of the machine; thus planers are on red card-board, drills on 
blue, and so on. The mounted machine-diagrams are then 
placed on a floor-plan of the factory and fastened into proper 
position (they can be shifted if the machines are shifted), and 
the routing of work is shown by ribbons slipped under the 
machines and stretched from machine to machine in the order 
of work. The flexibility of this ribbon idea commends 
itself.” 

It may be added that if several types of work were routed 
over this floor-plan, it might be advisable to use different 
colored ribbons to distinguish them. Furthermore, some idea 
of the volume of traffic or work along each route might be 
given by using ribbons of various widths, narrow ones for 
small or occasional work and wide ones for heavy trafic. The 
student will notice that, in accordance with the principles set 
forth below, the colors of machine-mounts should be in pale 
tints and those of work-routing ribbons in brilliant tints, to 
emphasize the route-chart over the floor-plan. 

When several floors are to be shown on the same map, that 
is, when the diagram must show in one plane a number of 
surfaces which are really not side by side, but one above the 


44 CHARTS AND GRAPHS 


other, the co-ordinate or cross-ruled chart-paper can be dis- 
; : re ites 

carded and a special form of paper, with “isometric ”rulings, 

used in its place. This paper projects without any perspective, 


Route OF;- 


~---—-- COMPUTING 
— CHARTS 
eoeecece -—DICTATION 


< 


ee ey 
ceeesiue ce ew ce ews eeeee 


o's 
ee 


ROUTE oF PARTS of REPORT 


Fig. 37. An Isometric Drawing. 


three dimensions in space, and following the general plan of 
charting on co-ordinate paper, can easily be plotted to show 
the various floors suspended one above the other. A full 
description of the principles of the paper will be given later, 
but the paper is noted here for its peculiar value in this type 
of composite chart. 

Such route-diagrams often furnish the most forceful way of 
presenting the weaknesses of a given arrangement of a factory. 
Traffic congestion is apparent through the number of crossing 
and confusing lines. A contemplated change which will result 
in a more orderly march of goods about the factory will appear 
upon such a chart with all its benefits made clear. A knowl- 
edge of the chart-form will therefore be useful whenever a 
revision or improvement of the lay-out of the plant is in view, 
with the object of shortening transportation distances or of 
installing ‘‘straight-line’’ processes. 


a 4 
4 
j 


COMPOSITE CHARTS 46 


Many other composite charts may be made besides, those 
showing motion through space. A classification-chart may be 


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Fig. 38. Routing on a Classification-chart. 
Showing by its shadings, the departments through which the circulation of goods 
and money takes place. 


superimposed upon a map. A route-chart may be superim- 
posed on a classification-chart. ‘Two classification-charts may 
be combined to show in a single chart both sets of logical 
relations. Route-charts themselves may be combined to show 
routings at different times or may be marked with distinctly 
classifying features, There is no limit to the variety of ways 


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COMPOSITE CHARTS 47 


in which the three principles of space, idea, and dynamic rela- 
tions may be interwoven on a chart. Nor is the number of 
possible strata or superimpositions necessarily limited to two. 
We may occasionally need three or four distinct strata, and 
indeed it is sometimes difficult to tell how many separate strata 
have been superimposed to effect a single chart. 

There are, however, very definite rules which it is well to 
follow in effecting the combinations. The first is that since 
two distinct conceptions of relations are being expressed 
through the same picture, the chart-maker should have clearly 
in mind their comparative importance, and be prepared to 
throw the emphasis in his construction upon the more im- 
portant one. Failure to do this will result in a chart in which 
the reader’s attention is distracted to the unimportant parts, 
by which it becomes hopelessly confused in trying to follow 
the important parts. The proper emphasis can be given by 
using in the superimposed chart, heavier shading or stronger 
lines which immediately distinguish it at a glance from the lines 
and shadings of the base-chart. Where color can be used, it 
should be used judiciously, after studying the effects of different 
colors to see which gives the right degree of prominence. 

The second rule is that since the superimposed chart gen- 
erally carries the message, the base-chart is usually one with 
which the reader is already supposed to be familiar. There- 
fore, when the base-chart, as well as the composite, is unfam- 
iliar to the reader, and particularly when a series of composite 
charts is to be shown on copies of the same base-chart, it is 
well to preface the composite charts with a single copy of the 
base-chart, with which the reader may first become acquainted. 
Then, when his attention is drawn to the composite, he will be 
able to use it at once with a full understanding of its significance. 

The conception of a basic or underlying pattern, as dis- 
tinguished from a resulting superadded pattern, is important 
and stays with us throughout the great majority of mathe- 
matical charts which follow. The term “held” will be used 
for this underlying pattern in these mathematical charts while 
the upper stratum of curves, bars, and other markings will be 
known as the “plotting”; and though the choice and construc- 
tion of the “field” will sometimes be found more important and 
often more difficult than that of the plotting, yet the field will 
always be suppressed or submerged by lighter lines and colors, 
to leave to the plotting its full significance. 


PART II. AMOUNT-OF-CHANGE ANALYSIS 


CHAPTER V 


STATISTICS 


Statistics is a word which has had an unfortunate history 
throughout its brief existence. To begin with, its pedigree is 
poor, for it is derived from old Latin words having nothing to 
do with its present meaning. When it was created a century 
or two ago, it meant “matters of State,” or in dictionary-ese, 
“matters pertaining to the State.” It was then used for 
those compilations of population, finance, and military strength 
which rulers liked tovyhave made about their various States. 
But it has travelled far from that meaning, until today in its 
proper sense it means any collection of figures and precise 
numerical information. 

The word was rapidly debased, until in common parlance a 
statistician was one who carried at the tip of his tongue a large 
assortment of appallingly uninteresting figures on widely 
irrelevant topics, which he seemed to have memorized from 
the encyclopedia with the sole purpose of boring us. Now 
figures are not'in themselves necessarily dry and dull—in fact 
the figures of your bank-account may be very engrossing to 


you. But figures on uninteresting subjects are a sure cure for | 


insomnia, to all of us. And it goes without saying that if the 
figures are not of consequence, the chart of these figures will 
deserve equally little attention. The point is that a chart is 
as weak as its own data, and a chart-maker must carefully 
weigh and consider his data before permitting himself the 
pleasure of illustrating them with a chart. 

But a worse charge than mere boredom is often levelled at 
statistics. It has crystallized into a familiar saying, “figures 
don’t lie, but liars figure.”” Mark Twain went so far as to 
remark at one time that there were three kinds of lies—namely, 
lies, damned lies, and statistics—wicked in the order of their 
naming. In short, the statistician is sometimes looked upon 
as one whose acquaintance with figures is so very intimate that 


48 


a 
"= 


STATISTICS ne 


he can readily take liberties with them, abuse them, present 
them in a false light, and deceive the layman. In this view 
he is little more than a common trickster, performing leger- 
demain with numbers, his magical results to be idly wondered 
at, but not to be trusted. And the moral thereof is clear, 
that he who would work with figures must be very, very sure 
that his figures are all correct, both in their computation, and 
their connotation. 

As a matter of fact, surprising as it may seem, we are all 
in the same boat, a whole nation-full of statisticians, in great 
or small degree. We all perform mathematical operations, 
arrange and study figures and precise data. We do it in our 
accounts, in our reports, our decisions, and sometimes in our 
sleep. If this be statistics, we are all guilty. And the odium 
which sometimes attaches to the word must be only superficial, 
for it does not attach to the practice or the subject-matter for 
which the word is a symbol. Surely we will not allow our- 
selves to be daunted by so empty a thing as a symbol or word. 
Let him who will, retain his shallow prejudice and throw this 
book aside here and now; and let the rest of us purge ourselves 
once and for all time of any lingering superstition against 
“statistics.” Let us resolve never again to utter a peep 
against the word or to be terrified by its use. 

There is, it is true, a more precise use of the word “‘statis- 
tics” in which a high degree of proficiency in handling masses 
of figures is presupposed. In this technical sense, the statis- 
tician is one whose ability to digest, compress, or extract 
significance from a multitude of related numerical data, is 
highly developed. The science of doing this is called by the 
queer name of “statistical methods.” In point of technical 
skill it stands somewhere between the science of accounting 
and the science of higher mathematics. If you have time to 
dip into it you will find it more interesting than either, because 
its applications are more varied than the former and more 
immediately practical than the latter. But for the purpose 
of chart-making or chart-reading, it 1s not necessary for you 


1 The notable books on the theory of statistics are: 
Bowley, A. L., Elements of Statistics. bit 
Yule, G. Udney, An Introductory to the Theory of Statistics. 
Shorter and more elementary texts are: 
Kelley, T. L., Statistical Method. AB 
King, Willford I., Elements of Statistical Method. 
Secrist, Horace, 4n Introduction. to Statistical Methods. 


ro CHARTS AND GRAPHS 


to have more than the usual grammar-school equipment in 
mathematics—or as much of it as you have not forgotten. | 

That the maker of charts must dabble in statistics 1s 
obvious; he who would build a house must first examine its 
foundations. But the chief requisite in examining your stat- 
istics is common-sense. You do not need deep mathematical 
skill wherewith to perform difficult mathematical acrobatics. 
Common-sense alone, without the aid of calculus or higher 
algebra, will enable most of us to understand a falling bank- 
balance, for example. Every school-child knows the meaning 
of a total, or an average, and all adult persons ought to, 
without regard for sex, color, or religious persuasion. Apart 
from such elementary understanding of the meaning of the 
language, the essential thing in ordinary statistical work is 
common-sense. If you have it, you can trafhe with figures 
safely, and will often recognize a condition without knowing 
the technical name or symbol for it, but if you haven’t it, all 
the mathematical skill in the world will merely befuddle you. 

Of course, for much work, or for continuous application to 
statistics, additional mathematical ability is an unquestion- 
able advantage. It is a good thing to know, for instance, that - 
there are several kinds of averages, each with a meaning all 
its own. It is well to have a nodding acquaintance with dis- 
persion, the word nodding here being used to denote familiarity, 
not sleepiness. And if you can shake a correlation coefficient 
by the hand it may help tremendously in a pinch. The man 
who can write an equation for his profits or his factory condi- 
tions, has the edge on the man who has to make lengthy tabu- 
lations. But the subject of this book is no more “how to be a 
mathematician” than “how to get common-sense,” and we 
will therefore drop both problems. We will split fifty-fifty on 
them, and in proceeding with charting of statistics, assume 
that the reader is gifted with common-sense, but not versed 
in higher mathematics. Where special need for certain mathe- 
matically technical terms or ideas arises, we will stop on the 
spot and explain these terms, but we will not build our mache. 
matical bridges until we come to them. 

One caution, and one caution only, in this chapter we Ge 
to make so clear that it will remain with the reader throughout 
the rest of the book. That caution is, do not be afraid to use 
your common-sense. In this matter an ounce of fore-sight is 
worth many pounds of hind-sight. The value of this advice 


STALISTICS 51 


will come to you through long and bitter experience, anyway, 
for it takes exceptional patience to scrap the results of weeks 
of research and start all over again at the beginning, just 
because of a failure to use common-sense beforehand. The 
most important time to bring your common-sense into play is 
before you lay pen to paper to take down a single figure. 
That is the time to ask yourself the all-important question, 
“What do I want to know?” Unless you can answer that word 
clearly, positively and concisely, you had better wait until you 
can, before doing anything else. 

“What do I want to know?” Write it down in black and 
white. Below it write the answer. Stand off and look at that 
answer as if you were a total stranger, and try to see whether 
it makes sense. When you finally have the question and its 
answer so clear that a child can understand, you will be ready 
to compile and investigate statistics to substitute for the 
answer. And when your work is finished, and you have boiled 
down immense quantities of figures and numbers to a simple 
coherent statement, see if that statement really answers your 
question. If it does, you have statistics which may be well 
worth illustrating with charts. If it does not, you had better 
forego the pleasure of making charts, for it will all probably 
have to be written off as a waste of time. 

You may think this all very simple and easy, but you will 
find it sometimes immensely difficult, and errors extremely 
costly. No rules can be laid down, for every case is a matter 
for individual study and analysis. But the consequences of a 
mis-step at this stage are grievous. And it is right here that 
so many amateur statisticians make their first mis-step. They 
become lost in the zest of hunting up and chasing down all the 
available related information; they allow themselves to be 
dragged off the scent of the fox by every jack-rabbit that 
crosses the trail. Red herring is their meat and in a shorter 
or longer time they bring home a mountain of “statistics,” all 
of which is “interesting if true,” but does not bear directly on 
the point. 

When you find this happening to you, you will be able to 
recognise it by the bewildered sensation in your solar plexus 
the first time a friend drops in and asks in a heartless way, 
“What’s the good of it?” And right there is a good time for 
you to stop—it would have been still better earlier—and ask 
yourself again, “What do I want to know?” Think back once 


52 CHARTS AND GRAPHS 


more to your original position and what you set out to learn. 
It is never too late to mend. If you find you are on the wrong 
track, bravely scrap the work you have done, though it hurts 
cruelly to do so, and strike out again for your goal. 

Your goal is a certain piece of information. Statistics are 
merely the road to that information. ‘The information itself 
will ultimately reduce to a comparatively simple statement, 
and if there are any statistics left in that statement, wipe them 
out by substituting illustrations or charts for them. The 
graphic method, whether used in the office to facilitate research 
work, or in the published report to facilitate understanding, 
should be confined to information which has value. It is 
therefore an obvious but extremely important rule not to 
begin a graph until the statistics have been carefully examined 
and their object or significance brought clearly into mind. 


CHAPTER VI 
WORK-SHEETS 


What would you think of a factory manager who kept his 
plant so cluttered up with raw materials, partly finished 
materials, by-products, and working machinery that his work- 
men had to climb over each other, and his materials had to 
be passed from operation to operation by long forward passes 
skilfully negotiated over ceiling-high piles of obstructions? 
Yet this is precisely what most of us do in the far more difficult 
case when our materials, workmen, and implements, are all 
intangible ideas, expressed by strokes of a pencil on paper. 
The old copy-book admonition to write clearly and neatly, 
may often save you from utter confusion and defeat, and will 
always greatly. speed your work. Straight-line computing 
methods are as important as straight-line factory methods.! 

Statistical data generally comes in the form of long columns 
‘of figures. If it is not already in this shape, you should so 
arrange it at once. It may be, for example, the reports of 
your sales in the various States of ‘the country. By listing 
the States in a column, you can write beside each one the figure 
of its sales, and your sales will then form a second column. 
Perhaps you wish to reduce these to per capita sales. In that 
case, beside the figure for the sales for each State, you can 
enter the population for each State, thus forming a third 
column of entries (a second column of figures). The per 
capita sales, which are merely the ratios between the two 
columns of figures, can then be entered still further to the 
right, forming a third column of figures (a fourth column in 
all). ; 

Frequently the computing is carried through a great many 
steps, each of which calls for one or more columns of figures. 


1 For an excellent discussion of the principles of tabulation, in addition to the 
works on statistical methods already referred to (page 49), see Edmund E. Day, 
“Standardization of the Construction of Statistical Tables,’ American Statistical 


Association Quarterly, March, 1920, p, 59. 
53 


$4 CHARTS AND GRAPHS 


DISTRIBUTION OF MsTAL MONXY IN THe WOKLD 
Approximate Stocks in Chief Countries, Dec. 31, 1918. 


TOTAL 


(Source: U. S. Statistical Abstract) 


¢ 
Dollars Population per 
cap 


10,127,084,000 1,529,179 ,000 


China ‘51,558,000 $36,042,000 
India 176,634,000 316,156,000° 
Russia 411,600,000 178,905,000 
United States 3,821,363 ,000 106,016,000 
643,672,000 67,810,000 
pean 482,646,000 65,965 ,60Q 
Austria Hungary 64,734,000 62,368,000, 
Dutoh East Indies 49,202,000 47,956,000 
Great Britain 122,561,000 48,089 ,000 
France 726,449,000 39,700,000 
Italy 249,137,000 36,646,000 
Brazil 43,690,000 26,542,000 
Turkey - 21,274,000 
Spain 658,851,000 20,600,000 
Korea 23,889,000 16,912,000 
Mexico 250,000,000 15,502,000 
Egypt 39,376,000 12,566,000 
Siam 41,582,000 8,266,000 
Cenada 191,827,000 6,075,000 
Argentina 8,066,000 
Belgium 56,806,000 7,658,000 
Rumania 1,000 7,508,000 
Netherlands 327,622,000 6,583,000 
South Africe 33,343,000 6,465,000 
Australasia 246,422,000 5,976,000 
Portugal 49,254,000 5,958,000 
Peru 32,691,000 5,800,000 
Sweden 88,856,000 6,713,000 
Colombia 10,768,000 5,071,000 

Morocco, French 24,638,000 

Serbia $3,486,000 

Ceylon 5,776,000 

Switzerlend 121,283,000 

34,092,000 

11,563,000 

62,649,000 

21,646,000 

44,911,000 

850,000 

4,140,000 

61,094,000 

Salvador 4,398,000 

Paraguay 482,000 

Dominican nepublie 800,000 

traits Settlements 17,263,000 


Nicaragua - 
Honduras - 
Costa Kica 2,112,000 
Ipxembourg 1,865,000 
British Honduras 168,000 


Fig. 40. A Simple Computing Sheet. 


The author has, he regrets to say, actually carried one investi- 
gation through so many consecutive steps that the resulting 
columns of figures, when pasted as close together as possible, 
side by side, without repeating any column, reached completely 
around the walls of an ordinary room. ‘There is generally no 
excuse for as much work as this, but it is well to bear in mind 
that every computing step may call for two or three columns 
of figures, and that if you are going to carry your figures 


through many steps, it will pay to have them in uniform 
columns, 


a, 


WORK-SHEETS > 5s 


Obviously the.arrangement of States or items in the columns 
should be standard throughout the work, so that any two 
columns can be readily compared. This makes the order of 
the items a matter for careful study. In listing the States of 
the union, for example, you have your choice of three arrange- 
ments. In the first place, you can list the States alphabetic- 
ally, which makes it easy for a stranger to find any particular 
State at once. Secondly, you can arrange them in the order 
of their importance (as viewed from the particular stand-point 
of your problem), which makes it easy for a stranger to focus 
his attention at once on the important States. But neither 
of these methods, though widely used, has anything more to 
recommend it than a certain possible convenience to strangers. 
For computing at least, the States should be arranged in a 
logical order, and a logical order is generally one that brings 
nearby States together, instead of scattering them about the 
list. The reason for this is that you may find you want to 
take off sub-totals (totals for East, North, South, and West) 
to get group-figures for groups of States. And if the States _ 
are already arranged or grouped together this is easily done. 
In fact, it is always well to carry these group-totals, because in 
checking through to locate errors they save much time. 

Having decided to arrange the States logically by territorial 
groups, you have next to decide how to group them. The 
census has one grouping, dividing the country into six or 
seven territories. But this grouping is not the best natural 
economic one, that is, it does not conform to natural business 
groupings. The Audit Bureau of Circulations has another, 
which is, for general business conditions, perhaps the best. 
But in most cases there will be individual factors which make 
it desirable to adopt a special arrangement of one’s own. 
Large sales organizations, for example, will already have set 
up their own sales districts and branch house territories, and 
in such cases it is best to make the grouping of States conform 
as much as possible to these. 1 

Of course, not all tabulations are tabulations of States by 
State figures. These figures illustrate, however, the principles 
of tabulating. We may have to work with figures, year by 
year through a series of years, or with month-by-month 
figures, through one year or more. In these cases, where we 
are working with divisions of time, the natural and proper 
way is to place the earliest periods at the top of the lise and the 


Alabama 
Arizona 
Arkansas 
California 
Colorado 
Connecticut 
Delaware 
Florida 
Georgia 
_ Idaho 
Illinois 
Indiana 
Iowa 
Kansas 
Kentucky 
Louisiana 
Maine 
Mary land 
Massachusetts 
Michigan 
Minnesota 
Mississtppi 
Missouri 
Montana 
Nebraska 
Nevada 
New Hampshire 
New Jersey 
New Mexico 
North Carolina 
_North Dakota 
Ohio 
Oklahoma 
Oregon 
Pennsylvania 
Rhode Island 
South Carolina 
South Dakota 
Tennessee 
Texas 
Utah 
Vermont 
Virginia 
Washington 
West Virginia 
Wisconsin 
Wyoming 


Fig. 41. 


CHARTS AND GRAPHS 


U.S. 
Census 


NEW ENGLAND” 
Maine 
New Hampshire 
Vermont 
Massachusetts 
Rhode Tsland 
Connecticut 


MIDDLE ATLANTIC 
New York 
Hew Jorsay 
Pennsylvania 


EAST NORTH CENTRAL 


Ohio 
Indiana 
Tllinois 
Michigan 
Wisconsin 


WEST NORTH CENTRAL 


Minnesota 
Towa 
Missouri 
North Dakota 
South Dakota 
Nebraska 
Kansas 


SOUTH ATLANTIC 
Delaware 
Maryland 


District of Columbia 


Virginia 

West Virginta 
North Carolina 
South Carolina 
Georgia 
Florida 


EAST SOUTH CENTRAL 


Kentucky 
Tennessee 
Alabama 
Mississippi 


WEST SOUTH CENTRAL 


Arkansas 
Louisiana 
Oklahoma 
Texas 


MOUNTAIN 
Montana 
Idaho 
Wyoming 
Colorado 
New Mexico 
Arixona 
Utah 
Nevada 


PACIFIC 
Washington 
Oregon 
California 


Audit 
Bureau 
of Cireulations 


NEW ENGLAND 
Maine 
Nen Hampshire 
Vermont 
Hassachusetts 
Rhode Island 
Conneoticnt 


NORTH ATLANTIC 
New York 
New Jersey 
Pennsylvania 
Delaware 
Maryland 
District of Col. 


SOUTH EASTERN 
Virginia 
North Carolina 
South Carolina 
Georgia 
Florida 


SOUTH WESTERN 
Kentucky 
West Virginia 
Tennessee 
Alabama 
Mississippt 
Louisiana 
Texas 
Oklahoma 
Arkansas 


MIDDLE STATES 
Ohio 
Indiana 
Illinois 
Michigan 
Wisconsin 
Minnesota 
Towa 
Missouri 
North Dakota 
South Dakota 
Nebraska 
Kansas 


WESTERN STATES 
Montana 
Wyoming 
Colorado 
New Mexico 
Arizona 
Utah 
Nevada 
Idaho 
Washington 
Oregon 
Oalifornia 


Red Cross 


NEW ENGLAND 
Maine 
Massachusetts 
New Hampshire 
Rhode Island 
Vermont 


ATLANTIC 
Connecticut 
New Jersey 
New York 


PENN. «DELAWARE 
Pennsylvania 
Delaware 


POTOMAC. 
Distr. of Col. 
Maryland 
Virginia 
West Virginia 


SOUTHERN 
Florida 
eorgia 
North Carolina 
South Carolina 
Tennessee 


LAKE 
Ind ana 
Kentucky 
Ohio 


CENTRAL 
IIlinois 
Towa 
Michigan 
Nebraska 
Wisconsin 


GULF 
Alabama 
Louisiana 
Mississippi 


NORTHERN 
Minnesota 
Montana 
North Dakota 
South Dakota 


SOUTHWESTERN 
Arkansas 
Kansas 
Missouri 
Oklahoma 
Texas 


MOUNTAIN 
Colorado 
New Mexico 
Utah 


Wyoming 


NORTHWESTERN 
Idaho 
Oregon 
Washington 


PACIFIC 
Arizona 
California 
Nevada 


Various Geographic Groupings of the States. 


WORK-SHEETS 57 


latest at the bottom, with possibly space for quarterly, half- 
yearly, or five-yearly totals, which ever may be desired. Still 
other types of items may occur, such as in a list of the various 
departments of a plant, the various salesmen of a selling 
organization, or the various products, and so on. 

The point is that whatever the items be with which we 
work, they should be arranged carefully at the outset, so that 
it will not be necessary to alter their arrangement later. They 
should be placed in a column if possible, so that the computing 
and tabulating can be made in parallel columns beside them. 
If the list is long, it should be broken up by blank spaces at 
convenient intervals, or better still, the items should be 
grouped together, for which sub-totals may be required, and 
blank spaces should be inserted between the groups for these 
sub-totals or part totals. But by all means, try to get the 
whole list on a single page, even at the cost of pasting additional 
sheets at the bottom, for the work will progress much faster 
on one large sheet which is complete than on several small 
sheets which are not complete. 

If much work is going to be done, it pays—and pays well— 
to have the printer rule up some sheets with the list of items 
printed at the edge of the sheet and the lines ruled in where 
they will be useful, horizontally from each item. It is a small 
matter, but worth noting, that the lines should be regular 
typewriter distance apart, so that should you wish to have 
figures typed on these sheets the typist can work rapidly and 
neatly, More important still, where adding machines are 


(Tone) 
NEW ENGLAND 
N. Hampshire, Mass., & Conn. 10,300 
MIDOLE ATLANTIC 


New York & New Jersey 2,600,000 
Pennsylvania 14,000,000 
SOUTH-EASTERN 
Maryland 524,000 
Virginia 429,000 
Alabama 2,390,000 
Tennessee 285 ,000 
Kentucky 772,000 


WORTH CENTRAL 
Ohio 8,530,000 


Indiana & Miohigan 2,940,000 


Tllihots 3,280,000 


WESTERN 
Towa, Mistourt, Colo,, Monts, & Ore. 465 ,000 


P10-IRON PRCDUCTION 
United Statee 
1920 


Fig. 42. An Incomplete State-list Geographically Arranged. 


58 CHARTS AND GRAPHS 


used, is to get the lines spaced in the same way that the 
adding machine prints the tape, so that figures do not need 
to be copied from the tape, but the tape can be pasted right 
on the sheet. In wide carriage machines, the tape can be 
dispensed with, and the figures printed directly by the adding 
machine on the sheets. These are devices to speed up the 
work, which are trivial in themselves but very important in 
their results, and, with a careful eye to your equipment, you 
will soon hit upon the most useful forms for doing your com- 
puting, standardise them, and call in the printer to prepare a 
number of blanks.? 


We have here considered carefully only the matter of the 
items which flank the left hand edge of your work-sheet forms. 
Technically, these items are called ‘“‘the stubs,” indicating that 
they are the labels attached to each horizontal line, or row of 
figures in the columns or column of the table. The whole 
sheet, filled with figures in orderly arrangement, is called a 
“table” or tabulation. ‘The vertical lines of figures are called . 
“columns” or sometimes “arrays,” while the horizontal lines 
of figures, that is, the sets of figures beside each stub, are called 
“rows” or lines. And at the top of each column of figures, 
the label which describes the column in the same way that 
the stubs describe the rows, is called a “heading” or “‘caption.”’ 


A whole chapter could be written about captions, or column 
headings. It isa fine art to make them at once clear and brief, 
and to arrange them so logically that the thought moves easily 
from one heading to another. In most cases of several columns, 
two or more captions can be bracketed together by a third 
common group-caption. In this way, the headings often take 
on the form of miniature classification-charts. The best prac- 
tice is to box in the headings carefully, so that they will be 
clearly understood. In work sheets they should be arranged 


> The position of the total (or sum) in a tabulation is an important matter. There 
are two possible positions: first, at the beginning or top of the table; second, at the 
end or bottom of the table. The first is the statistical position; it is correct for a 
published table, as it places the most important item, the total, or whole, first 
before the reader’s eye. When parts are themselves further subdivided, their totals 
should also be placed before their parts. The details, or parts, can be in every case 
further indented than the totals. This practice should be adhered to in charts and 
in published or recorded tables. 

The second position, at the end of the table, is the accounting position. It is 
correct for all cases in which the work of summing up the parts must be frequently 
undertaken. Needless to say, it is essential for forms to be used in adding and listing 
machines, and is advisable in general for work-sheets. 


WORK-SHEETS 59 


ILLITERACY IN THE UNITED STATES 
Illiterate percentage of each olass (by age, sex, race) of the populatiom 
for each group of states, 1920 
(Soures:- U. 8S. Consus) 


Totel population over 10 years of age. 


(Over 21 years) | Foreign 
Native born 

[| wete | rome. [| wsite | waite | nogro 
200 18.1 22.9 


7.0 


TOTAL U. &. 


Hew England 6.0 7.1 


Middle Atlantio 


East North Central’ 


Weat North Central 


South Atlantic 


East South Central 


West South Central 


Mountain 


Paoific 


Fig. 43. Classified Headings to the Columns. 


in the order in which they will be computed, so that the work 
of computing moves as much as possible to the right, always 
preferably deriving each column from the immediately pre- 
ceding columns. Where two or more columns, which are in 
themselves the results of several columns, must be combined, 
each can be left at the right-hand edge of separate computing 
sheets and then by cutting off the remaining paper, or by fold- 
ing it back, you can lay the sheets one over the other so that 
the columns to be worked over will ‘appear side by side. 

One of the cardinal rules for computing tabulations—and 
it applies only in lesser degree to final tabulations ready for 
publication, presentation, or study—is that every column head 
should include a number or letter identifying the column. 
This rule has even been extended by some authorities to the 
stubs as well. The advantage of the number or letter is that 
it makes reference to the particular column very easy, either 
in texts, notes, conversation, or formulae. 

In addition to a symbol for the column, you should also 
have in your column-head, a note explaining the source of the 
figures in the column. This note can either be the name of 
the authority from which the figures were copied, or it can be 
the formula by which the figures were computed from other 
columns. It will save you much trouble in correcting the 
errors of computing clerks, and much time later on when you 
come to refer to the sheets and do not remember the various 


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WORK-SHEETS 61 


steps clearly. For the clerks who are doing the computing, 
this note is a standing bill of instructions. And the time will 
SAVINGS BANK STATISTICS 
Number of savinge banks and savings bank depositore; total, average, and percapite 
deposits; and ratios between banks, depositors, and population as specified below. 


United States, 1820-1920 
Arranged from U. S. Statisticel Abstract 


rons Per- | Depositors per 
Banke Deposits Depositors dopedit Population pba A ae ee 


hunter | Dollera 


Yeer 


1820 ee 1,138,876 8,635 | 132 9,638,453 

1625 16 2,537,082 16,931 11,150,000 144 ,000 
1830 36 6,973,304 38,035 12,866 ,020 357,000 
1835 $2 10,613, 726 60,058 14,710,000 283 ,000 
1640 61 14,051,520 78,701 17,069,453 280,000 
1845 70 24,606,677 145 ,206 19,970,000 285 ,000 
1850 43,431,130 251,354 23,191,876 215,000 
1665 84,290,076 431,602 27,256,000 126,700 
1660 149,277,604 693,870 31,443,321 113,100 
1665 242,619,382 980,844 34,748,000 109,600 
1670 549,874,358 | 1,630,846 38,658,271 74,500 
1675 924,037,304 | 2,359,664 43,951,000 56,900 
1880 819,106,973 | 2,335,582 50,155, 783 79 ,800 
1885 1,095,172,147 | 3,071,495 56,148,000 87,000 
1890 1,524,644,606 | 4,258,893 63,056,438 68,400 
1895 1,810,597,023 | 4,875,519 69,579,868 68,400 
1900 2,449,547,885 | 6,107,083 76,129,808 76,100 
1905 3,261,236,119 | 7,696,229 84,219,378 68,100 
1910 4,070,486 ,246 | 9,142,908 92,267,080 52,600 
1916 4,997,706,013 | 11,285,755 99,342,625 46,000 


11,437,556 106,418,175 62,300 


6,636 ,470,000 


Fig. 45. Column Symbols and Computing Instructions. 


come when you will map out your work in this way, merely 
filling out stubs and the column-headings yourself, and leaving 
the entire computing task to clerks. 

It is also just a question of time before you will come to 
look upon these tabulated figures as so many various descrip- 
tions or phases of the original set of stubs. The column- 
headings tell you the type of description or phase, but in the 
end the stubs are the basis, and the figures are derived from 
them. Mathematicians have a word “function” for such rela- 
tions. Using that word, we would say that the tabulated 


do CHARTS AND GRAPHS 


figures are functions of the stub-figures or items, meaning 
that their values are derived therefrom. Glancing across the 
various lines, you will see that all the figures on the same line 
with a stub are but various and varying functions of that stub, 
the symbol at the top of the column identifying the function 
and the caption describing it. 

Glancing down any column, you will see that the figures 
change from item to item. They form a series of varying 
values. Each column contains the various values of the func- 
tion described by the caption head. Each column can be 
looked upon as a series of figures which are the readings or 
values of the item, described by the column head. And the 
point is that while the stubs, and captions, were independently 
arranged by yourself, the figures in each column are derived 
from or attributed to them and so are dependent upon the 
stubs and captions. In short, while the stub is an “independ- 
ent variable,” the series of figures in the other columns are all 
“dependent variables” with regard to the stub. 

Sometimes the independent variable is called the “x- 
variable’ and the dependent variable the “‘y-variable.”’ In 
that sense the values of ““y’’ will all depend upon the values or 
meanings of ‘‘x.”’ We might go so far as to label the column 
of stubs “‘x’’ and the captions “‘y,”’ being various kinds of ‘‘y”’ 
variables. Think of the first one as “‘y,”’ the second column 
as ““y,,” the third as “‘y.,” the fourth as “ys,” and so on, and 
it will always be clear to you that the stub is the independent: 
or x-variable and the other columns are the dependent or 
y-vatiables. A second table on an entirely different aspect of 
the same items or stubs, could be called a “‘z-variable,” meaning 
that while it was also a dependent, it was distinct from the 
first set of dependents. Asa matter of fact, this is all relative, 
for at some time you may treat one of the columns of figures 
as an independent variable and the stub as its dependent. 
‘But it is useful to begin thinking of your tabulation as having 
both independent and dependent variables contained in it. 


CuaptTer Vil 
CO-ORDINATES 


The better to demonstrate their simplicity, we have dis- 
cussed in the very first chapter those few technical terms which 
will be essential to the student in this elementary work. In 
that chapter the analogy is drawn between the checker-board 
arrangement of streets in certain American cities and the criss- 
cross rulings of most chart paper. In this chapter this par- 
ticular type of ruling will be carefully re-considered for the 
benefit of those whose zeal in the subject has led them to skip 
the first chapter, and a few other forms of ruling will also be 
touched upon with their relations to this fundamental system. 
The chapter will not be interesting reading, but it deserves 
close study in order that the remainder of the subject may be 
clearly understood. 

Station yourself in an open field where you can, without 
difficulty, move in any direction—even upward with the aid 
of an aeroplane or downward with the aid of a good fast shovel. 
At the point where you stand, drive a stake into the ground 
and consider it your base of operations for all other points on, 
above, or below the field, that is, the point from which you 
can measure their distances. We will call it the “point of 
reference”’ or “‘point of origin” and mark it “O,” which accord- 
ing to your taste may either stand for the word “origin” or 
for its zero distance from the origin. 

Facing in any direction at this point walk forward a dis- 
tance, let us say, of five steps, in a straight line. It 1s obvious 


that you could walk forward indefinitely in this direction, 


always reaching a greater distance from the starting-point “O,” 


fr ncn eeepc 
i?) 1 2 3 4 5 
Fig. 46. 


and that you could always measure this distance by stakes at 
each foot-step numbered “1,” “2,” “3” and so on successively 


63 


64 CHARTS AND GRAPHS 


as you pass them. You could then instantly locate any point 
along your path by the number of the stake. Points midway 
between full steps can be given fractional values. Such points 
would be definitely and unmistakably identified if, in addition 
to telling their distance from the origin, you also tell the direc- 
tion in which you had walked when you left the stake to 
measure them. Calling this direction ‘‘x,”’ you would specify 
the points completely by calling them ‘‘x, 1,” “x, 2,” “x,3,” and 
so on. 

Having walked forward, however, only five steps, you 
would reach the point “‘x, 5. ” Suppose here you stop and begin 
to retrace your steps, walking backwards. You would notice 
that now instead of the distances from “O”’ increasing, they 
decrease as you walk backwards, until after five backward 
steps you are again at the zero-point. But continue to walk 
backward and you will have again the phenomenon of increas- 
ing distances from this point as your steps increase. In other 
words, ‘along the same straight line, there are two sets of 
distances mirroring each other at the zero-point. If you walk 
backward ten steps all told, from the point “x, 5,”’you cometo 
another spot which is also five steps from the origin and in the 
same straight line. There must be some way to distinguish 
the two and to distinguish all the other pairs of distances in 
this straight line which we have called ‘“‘x.”’ Suppose that for 
this purpose every distance reached by walking forward from 
the origin be called positive and every distance reached by 


Fig. 47. 


walking backward from the origin be called negative. Leaving 
the first set of stakes as before marked ‘x, 1” (or “x, +17”), 


“¥, 2° (or “x, “2” >and) so“on, you can similarly mark the 
stakes at your backward a from the origin, by numbering 
them “x, -1,” “x, -2,” “x, -3” and soon. In this way you 


will Bue enary every es along the straight line, both 
on one side of the origin (in the ‘“‘x’’ direction) and on the 
other side (in the “‘ —x’’ direction). 

Here you have set up measurement in one direction, that 
is, along one “dimension.” ‘There is a technical name for this 
“direction” or “dimension,” which it will pay you to learn, 
namely “axis,” In fact, we can already speak of it as the 


CO-ORDINATES 65 


“x-axis,” or axis of “x” measurements. But leaving aside 
technical terms, your common-sense will tell you that you 
have set up the only kind of measurement possible in one 
dimension, namely “linear measurement.” In this particular 
case, your foot-step—one half of your pacing distance—is 
your “unit of measurement.”’ Any other unit could have been 
taken. You might have laid a certain stick down repeatedly 
and marked off the number of times it could be laid end over 
end. If you had a number of sticks of the same length you 
could lay them end to end and leave them lying to form one 
long pole or rod cut into equal parts. But whatever the 
length of your unit of measurement may be, you will, by num- 
bering the units successively in both positive and negative 
directions, have graduated or “calibrated” that one long im- 
aginary rod with a “‘scale’”—the scale or calibrations being the 
numbers or countings from the zero-point in both directions. 

It will occur to you, however, that while you have an excel- 
lent system for locating points in that one line along which 
this imaginary rod lies, that is, along which you walked, you 
have no means of identifying points elsewhere in the field. 
Suppose, therefore, you return to the “origin”? with a short 
actual rod which is long enough to reach from point “x, 5” to 
point ‘‘x,—5” and lay it actually between those two points. 
(You can extend this rod in your imagination indefinitely, in 
both directions, but a rod of ten steps length is handier to 
carry about than an indefinitely long one.) At every stake 
mark the corresponding point on your rod, with the same 
numbers, so that you can dispense with the stakes entirely, 
and with the rod in this position you can still locate points 
along the “‘x’’ direction from’ the origin, either positively or 
negatively, at once. 

Now stand at the origin with the positive markings on the 
rod to your right and its negative markings on your left hand, 
and you will find yourself facing in a direction at right angles 
to vour first line of walk. Strike out in this new direction. 
As you walk you can again keep track of the distance by count- 
ing foot-steps, and you will again in this new direction find 
both positive and negative values which mirror each other. 
You can find positive values by walking forward and negative 
ones by walking backward from the origin. This new direc- 
tion, lying at right-angles to the original “x” direction, you 
can call the “‘y” direction, and specify points along it as “‘y, 1, 


66 CHARTS AND GRAPHS 


ny Z; ” Pgs ee and “ny, es ‘yy, Shes +) Ws oe ”) and so on. 
And so you will have set up a second scale at right angles to 
°5 


4 


Fig. 48. 


your first or “‘x-scale,” by means of which you can identify 
points along the “‘y’’ direction from your origin. The unit of 
measurement may pe may not be the same as the unit used in 
the first measurement, but of course whatever unit you adopt 
in the ‘“y-scale’”? must be adhered to as closely therein as was 
the “‘x-unit”’ along the ‘‘x-axis.”” Being in a new direction, 
the distances have no relation to those in the old direction, 
but it is obvious that along either direction the units therein 
must be uniform. And since we called the first direction the 
“x-axis,” we may call this new one, at right angles to it, the 
““yeaxis.”” 

Now if you will pick up the rod, lying in the “‘x-direction”’ 
and carry it with you as you walk in the “y-direction,” without 
swinging it or moving one end faster than the other, but taking 
care to hold it rigidly as you walk, you will see that every 
point along the rod, marking distances along the ‘‘x-axis” 
describes a straight line parallel to the “y-axis” in which you 
walk. Here you will have a means for igarialag ie any ‘point 
on the field. You need but to carry the rod or x-axis out 
along the y-axis until the rod crosses the point you wish to 
identify. Then note the point on the rod with which it co- 
incides, such as “‘x, 3,” and the point on the y-axis to which 


; 


_CO-ORDINATES . 67 


you carried the rod, such as “y, 6.” You will define the point 
as being ‘‘x,3; y, 6,” and you will search in vain for any other 


ada 


“ 9-4 8 2 ef (9) 7 2 8 é 6 
Fig. 49. 


point which can be similarly described. Three other points 
you will find which have the same numbers, but not the same 
signs; these points will be “x, 3;.y, —6; “x, -3; y, 6” and 
“x, -3; y, -6.” You will find that your two axes cut the 
field into four quarters, in each of which all points have the 
same combination of signs. 

If you do not wish to carry the rod back and forth along 
the y-axis, or a similarly marked rod in the y-axis back and 
forth along the x-axis, you can lay a series of rods parallel to 
each other and perpendicular to the x-axis, crossing that axis 
at each point marked off on it, and another set of perpendiculars 
along the y-axis so that your entire field is crossed by these 
measuring rods in two directions. Thereafter you can locate 
any point on the field by walking out either axis and then 
turning at the right distance and following the perpendicular 
there, parallel to the other axis.! 

You now have at your disposal a means for identifying any 
point or any number of points upon a given field, or in precise 

1 The co-ordinate axes need not be at right angles to each other, they may be drawn 


at any other angles desired; in the latter case, they may be thought of as really at 
right angles, but seen from a side rather than a direct view. 


68 CHARTS AND GRAPHS 


language, in a plane surface. A plane has, as geometricians 
say, two dimensions, commonly called length and breadth. 


“4 +8 +2 -1 0 4 & 
Fig. 50. Field with Equal Scales. 


For these you have laid down two straight lines, or axes, at 
right angles to each other. One of these you have called the 
‘x-axis,’ the other the “‘y-axis.”” Both have been calibrated 
or measured off and scales attached. And with this simple 
mechanism you can take measurements of various objects in 
your field and record locations so precisely that others can, 
by your records, be led to the very same objects, or can dis- 
cover without possibility of doubt, the exact spots upon which 
the objects had been placed when you measured them. In 


See ee aor ee ee ee ei aS 


Fig. 51, Scales of Axes Unequal, 


CO-ORDINATES @ 


short you have the means for identifying any point upon a 
plane surface. 

It is a fortunate coincidence that the paper upon which we 
ordinarily write is flat and its surface can be considered a plane 
surface. For this enables us to apply a reverse English to our 
measuring device, and use it in making precise illustrations of 
such fields as we have measured. Now instead of being given 
a field with objects and being required to set up the measuring 
device in order to ascertain the location of the objects, we will 
be given paper already ruled off with the measuring device, 
and will be required to place thereon indications of the objects. 

The paper ruled off in this way is called “co-ordinate paper” 
for reasons which will presently appear. Its rulings represent 
a series of parallel lines laid crosswise over another series of 
parallels, and we are at liberty to select any two intersecting 
lines for our axes and mark off our scales or measurements 
along these lines as large or as small as may suit us. And you 
will find that more than half the charts you ordinarily encounter 
will be constructed in this way. The horizontal lines are called 
abscissae, the vertical ones ordinates.2 Taken together, these 
co-ordinate rulings form what is, in chart-making, technically 
called the “field” of the chart, being the background upon 
which the distinctive portion of the chart is superimposed. 
And as will be remembered, from the chapter on superimposi- 
tions, the basic portion or field should be as unobtrusive as 
possible, that the important features may receive more atten- 
tion. The field should always be drawn lightly, with thin 
lines, and with no more co-ordinates ruled in than are necessary 
to afford the chart-reader ease of comparison. If possible, 
the field should be in green or grey ink, as this further sub- 
merges it. 

The origin or zero point we have so far taken within the 
field, so that the field is cut into four quadrants in which the 


2In consequence of that quaint genius for unnecessary trouble sometimes exhib- 
ited, a very serious discussion has occasionally arisen as to whether the abscissae 
should be ruled upon the paper with their upper or lower edges upon the exact positions 
which the lines signify. The idea of those who favor the upper edge appears to be 
that the abscissae are shelves upon or above which plotted points rest, when permitted 
to do so. The idea of those who favor the lower edge is not so cogent. Curiously 
enough, similar debate has never raged around the position of the ordinates. As a 
matter of fact, few charts are so precisely drawn or so finely adjusted that the thick- 
ness of the ruled line is material, and in every case, the obvious place for all ruled 
lines is a centered position, that is, one in which their two edges are equidistant from 
the precise desired positions of the imaginary lines they represent. 


70 CHARTS AND GRAPHS 


“3 +2 1 0 1 2 3 4 5 6 ? 
iz. 52. Origin of Chart near One Edge of Field. 
values of points mirror and repeat themselves with different 


plus and minus signs. But in practice a large proportion of 
our measurements or data present no negative values, at least 


Fig. 53. Origin in Corner of Field. 


along the x-dimension. The right-hand upper quadrant, or at 
most the two right-hand quadrants are then sufficient, and we 
can omit the remaining quadrants entirely. As a result, the 
ordinary chart on these co-ordinate rulings shows the value of 
zero or its origin-point along its left hand edge and generally 
in the lower left-hand corner. The method is still the same, 
but a portion of the field is merely being omitted, because 
useless. In fact, we can even go further and begin the chart 


CO-ORDINATES “1 


Fig. 54. Origin Not Shown in Chart. 


out to the right of the origin, omitting the origin itself when 
that also is not to be used. 

This system of parallel lines to two axes, which are them- 
selves at right angles to each other, belongs to what is called the 


aegoemewa 
ano Soy ey yee aay 
prea raina 
Panay me a) Te} 
ils die EBS) el SS aT 
CE Et a 
o4 23 #2 of 0 Z Fe LO eee 6 
Fig. 55. Field with Co-ordinates Not Perpendicular. 


Cartesian system of co-ordinates. That system need not stop 
with plane two-dimensional surfaces but can be extended to 
cover three-dimensional space, by the very simple expedient 
of perpendiculars erected at the intersections of the parallels. 
You may think of this as lifting your entire net-work of rods 
up over your head as you stand on the “origin” in your held, 
or forcing it down into the earth below you; calling the vertical 
line through “O,” the “z-axis” and measuring upward distances 
positively and downward distances negatively. But because 
three dimensions are not easily represented on a flat piece of 


72 | CHARTS AND GRAPHS 


paper, you will find few charts which have at the same time 
three axes. 


Fig. 56. The Three Axes of Three-dimensional System of 
Perpendicular Co-ordinates. 


Students of trigonometry will recall another method of 
measuring. They will say, standing at your point of origin 
with the “‘x-axis” rod in your hands, do not walk forward, but 
turn slowly around. ‘The points on the rod will then describe 
a circular movement about you, and you can locate any point 


Lt] 
St 
WZ 


Fig. 57. Polar Co-ordinates. 


in the field by noting the distance on the rod and the angle 
through which it has been turned. Indeed, you will find many 
a chart which is built on this principle. Later on when we come 
to some examples of it, we will have to explain its peculiar 
qualities. For the present it is enough to say that the method 
of “‘polar co-ordinates” can be used. 


CO-ORDINATES 73 


The great mass of chart work is built along plain co- 
ordinates. The simplest form of all uses but one axis, and 
consists only of a straight line; the commonest form uses two 
axes, and consists of square or rectangular outlines. Occa- 
sionally we have use for three axes, using three dimensions, 
and we must not forget also, the circular two-dimensional 
device with polar co-ordinates. 

_ All this is so extremely simple that we hesitate to dwell so 
long upon it. But unfamiliar names are great bug-bears; let 
the idea be as simple as pie, technicians will come along and 
give it a long high-sounding name, generally created on the 
spur of the moment by themselves out of ancient Latin or 
Greek dictionaries, but which even the old Romans and 
Athenians themselves would have been unable to understand. 
We must not let them fool us with these long empty names. 
On the contrary, when we see ‘‘co-ordinates’”’ we will know 
that it means nothing more than criss-cross lines. We will 
remember the criss-cross checker-board arrangement of roads 
in American cities, and unless we hail from Boston, we will 
think of “‘street’’ when we see the word “‘abscissa” and we will 
think of “avenue” when we see the word “ordinate.” The 
“x-axis” is ‘“Main Street”? from which on both sides the other 
streets (abscissae) are counted; the “‘y-axis”’ is “Main Avenue”’ 
from which on both sides the avenues (ordinates) are counted. 
The point is that under no consideration will we let ourselves 
get disturbed by the unfamiliar names, and in a little while 
we shall be able to swing these words around with the best of 
them. 


3 “Main Avenue” (the ‘‘y-axis’’) is crossed or cut by the streets (abscissae— 
Latin for cut-offs) and ‘‘Main Street’’ (the ‘‘x-axis’’) is crossed or cut by avenues 
(ordinates—their Latin failed them here—ordinates means arranged in order). More 
than that, if the streets run east and west, and the avenues north and south, then 
positive x is east along a street from Main Avenue and negative x 1s west; positive y 
is north along an avenue from Main Street and negative y is south. Upper Central 
Park West, for example, is negative x from Fifth (Main) Avenue and positive y from 


Battery Park, 


Cuapter VIII 
DIMENSIONS AND VARIABLES 


While it is deemed essential for artists to understand the 
nature of crayons, brushes, and like materials for their 
work, yet we often observe that they make a great to-do over 
their study of human anatomy, landscape scenery, and the 
subject-matter of their work in general. In the same way we 
who would illustrate mathematical facts must, it is true, mind 
our “‘p’s” and ‘“q’s’—which is to say, our “abscissas and 
ordinates,’ but these are merely the tools with which we work, 
and most of our attention should be directed to our subject- 
- matter—the data or statistics we wish to illustrate. The closer 
is our analysis and understanding of the nature of this subject- 
matter, the clearer and more accurate will be our graph or 
illustration of it. 

Because this book is largely a manual on the craftsmanship 
of charting, and will therefore be mostly devoted to the exam- 
ination of the various types and kinds of charts with which 
you can illustrate statistical facts, we here take one last occa- 
sion, before the curtain rises on the charts themselves, to appeal 
to the reader, on behalf of his own common-sense, and urge 
him to review carefully his analysis of his statistics, before 
laying ruler to chart. 

We make this appeal at this time when the reader has 
finished his statistical work and is addressing himself to its 
charting. This is the time when, once more, he must stand off 
and scrutinize the statistics which he has compiled in the face 
of such great difficulties—the statistics which are as dear to 
him as his own thought-children—and scrutinize them with 
cold, calculating, dispassionate eyes. Do not make the mistake 
of plunging into the charting-work direct from the statistical 
work, but once more carefully weigh those statistics against 
the original question, ‘““What do I want to know?” 


74 


DIMENSIONS AND VARIABLES sigs 


The reason for this is two-fold. In the first place, this final 
examination will give you valuable suggestions as to what part 
of the statistics the chart should emphasize. It will enable 
you to eliminate from your chart entirely what is not significant 
to your inquiry, though the same may be necessarily retained 
in the table for reference. It will enable you to focus the chart 
upon the essential information and present it to others in pre- 
‘cisely the light or relation in which you wish it seen.! In short, 
it will take the “straw” out of your graph and leave only the 
wheat. 

In the second place, you need this last-minute scrutiny of 
your figures, or some similar analysis when your results have 
come clearly into view, to determine how you will chart the 
statistics. Though it will not always decide the precise form 
of chart for you, it will settle the fundamental principles of 
that chart, and you can easily modify details later. The re- 
mainder of this book will be devoted to the details of the 
various charts, but here and now let us attack the fundamental 
principles. 

You have seen in the last chapter that measurement of 
space is based upon linear distance in each of its dimensions. 
A one-dimensional space would require but one axis or direc- 
tion for measurements. A two-dimensional space would 
require measurement in two axes or directions. A three- 
dimensional space requires measurement in three axes or 
directions. In short, there must be as many axes for measure- 
ment as there are dimensions. 

Now in precisely the same way, your statistical data can 
be considered as having one, two, or more dimensions. If 
there is but one dimension in your data, you should use but 
one axis or direction on the chart-paper to picture it. If there 
are two dimensions in your data you can use several one- 
dimension charts or one two-dimension chart. If there are 
three dimensions in your data, you can either present them 
two at a time in several two-dimension charts, or try your 
hand at one three-dimensional chart—a more complicated 


form. 


1 It is not intended here to suggest that it is possible or desirable to practice 
deception, with any correctly-made chart, but only that the chart-maker can bring 
out various aspects of the truth about his data with greater or less clearness, by his 


choice of charting method, 


76 CHARTS AND GRAPHS 


How can you tell how many dimensions there are in your 
data? You can tell by the form which your data takes when 
you have neatly tabulated it. You must allow one more 
dimension to your chart than the figures to be plotted actually 
require for correct tabulation. That’s the rule. For a figure 
itself must be considered as having one dimension. (Imagine 
each figure as reaching up perpendicularly off of the paper and 
this will become clear to you.) 

This rule can therefore be stated in another way. For 
single figures use one dimension or axis on your chart. Fora 
series of figures, arranged either downward in a column or 
across in a row, use two dimensions. For a series of series of 
figures, comprising a row or line of columns side by side, use 
three dimensions. These last cases are not frequent, and are 
limited to occasions where you could have presented the figures 
in each row in a two-dimension chart, or the figures in each 
column in a two-dimension chart, but wish instead to show the 
figures by columns and rows simultaneously, requiring, there- 
fore, a three-dimension chart. 1 

It is always possible to present several charts in one. For 
instance by using a number of one-dimension charts, you may 
adhere to the one dimensional principle, but show several at 
the same time. A series of horizontal bars, for instance, is an 
illustration of this. (But when the series is arranged with care 
as to the downward axis also, these many one-dimension charts 
can sometimes be made into one two-dimension chart.) Or 
you may use a number of two-dimension charts, thereby 
adhering to the two-dimensional principle, but showing sev- 
eral at one time. A number of curves on the same chart-field 
illustrate this. However, where the base-lines are differenti- 
ated with care, these can often be made into one three-dimen- 
sional chart. Charts into which several single charts have been 
compressed are called multiple charts. 

This brings us to an important. exception or modification 
of the simple rule for dimensions. For it is necessary that you 
distinguish between variables and other functions. A table 
which includes two or more functions which are not variables 
is really not a simple table but a multiple table, into which 
several simple tables have been combined. Thus, if the stubs 
of your tabulation (that is, the items at the left of each line) 
form a variable, they should be counted as requiring a dimen- 
sion on the chart. But if they are not values of a mathe- 


DIMENSIONS AND VARIABLES a} 


matical variable, they do not add to the dimensions of the chart 
but merely form a list of the number of charts which may or 
may not be compressed into one multiple chart. The same 
considerations hold true of the “column-headings” or “cap- 
tions” of the tabulation. The stubs or headings constitute 
variables, when their arrangement is fixed by their mathe- 
matical sequence, but are not variables when they may be 
freely shifted about. 

The question of whether you are plotting individual figures 
(no variable, one dimension), or series of figures (one variable, 
two dimensions), or series of series of figures (two variables 
and three dimensions), is therefore a matter of whether the 
arrangement of the component partsis fixed or not. If your 
individual figures (together with their stubs) can be shifted 
freely up or down in their places, you are only plotting several 
single figures, and need to use but one dimension. If however, 
their arrangement is fixed and dependent on each other, you 
are plotting one series of figures and need two dimensions. 
Likewise if your columns or rows of figures can be freely shifted 
about among each other, you are plotting several series and 
need only two dimensions, but if their arrangement is fixed 
and dependent upon each other, you are plotting a series of 
series and need three dimensions. 

All of this may seem very confusing just at present, but as 
time goes by and you become more familiar with the nature of 
your figures, you will begin to understand the interrelation of 
these figures and then you will be able to use the rule just 
considered to determine quite arbitrarily in advance the type 
of chart you will need for illustrating them to the best advan- 
tage. We will therefore table this matter for the present, with 
the purpose of returning to it later on with a fuller under- 
standing. 

A corollary of this rule, however, you can keep and make 
full use of from the outset. That is, never to use more dimen- 
sions than are necessary for your chart. Do not use two 
dimensions when one will serve yout purpose, nor three when 
two will do. If, through an excess of zeal, you violate this rule, 
and use more materials than are necessary, you will merely 
defeat your purpose and confuse the reader of your chart. 

Let us take a simple example. Suppose we wish to illus- 
trate with a chart the relative sizes of two cities, with popula- 
tions 5,000 and 10,000 respectively. Obviously the second 


78 CHARTS AND GRAPHS 


town is twice as large as the first. There is but one variable 
present (none in the data-stubs or column-headings), the sizes 
of the cities. According to our rule we will use one-dimension 
charts for these figures, but as there are two of them, we will 
have to use two charts. 

Because the charts are single-dimension or single axis ones, 
it is obvious that they will be straight-line ones. Two lines or 
bars, the one of which is twice as long as the other, will illus- 


| 


Fig. 58. 
trate the two populations, so that the reader cannot help 
arriving at the conclusion that one city is twice as large as the 
other, merely from a glance at the chart. This chart will be 
both clear and accurate. It is therefore correct. 

But suppose we had in this case used two-dimension charts. 
Suppose for example that we had used square areas instead of 
lines. And because the second city is twice as large as the 
first, suppose we therefore made each dimension in the second 


rags 


Fig. 59. 
chart twice as great as the corresponding dimension in the 
first chart. You will already be objecting that the area of the 
second chart will not be twice as great as the area of the first, 
but four times as great. Look at the chart. Notice how sur- 
prising is the difference between the two cities as here repre- 
sented. 

And notice also that it is very hard, from the chart alone, 
to decide exactly what the ratio between the two areas is. The 
eye does not readily compare areas with precision. At first 
glance one might be inclined to say that the second area is 
five times as great as the first; though we who made it know 
that it is only four times as great. This illustrates a very 
important rule—that it is difficult to compare areas. That 
rule follows from the original principle, not to use more dimen- 
sions than necessary, but it is worth keeping in mind as a 
particularly important phase of the principle. 


DIMENSIONS AND VARIABLES 79 


. But you will say, “Cannot we show the areas in true pro- 
portion 2” We can, if we will but take the trouble. A little 
figuring will show you that if the areas are to be in the ratio 
of 5,000 :10,000 or 1:2, the sides of the squares must be in the 
ratio of one to the square root of two. Get out your pencil 
and paper—or your slide rule—and figure out the square root. 
It happens to be 1.414. Consequently, if the square areas are 
to be in the proportion of one to two, the sides of the squares 


Pisin 


Fig. 60. 


must be in the proportion of 1 to about 1.4. Now draw the 
two square charts in such a way that the sides of the second 
are 1.4 times as long as the sides of the first. | 

A glance at the resulting chart will disappoint you. Instead 
of making the difference in size clearer, you have apparently 
minimized it. The two areas no longer seem so very different 
in size. It will take a clever reader indeed to realize from a 
study of the two areas that the second city is fully twice as 
great as the first city. And the worst of it is that many 
readers will, through ignorance, fall back on the length of 
the sides as a basis of judgment, and decide that one city is 
only half as large again as the other. In short whether you use 
the sides or the areas as your own basis, there is a good chance 
that your reader will happen to use the other basis and so be 
entirely misled by your chart. 

Obviously this is no less true when we use circles or other 
regular areas in the place of squares. A circular, like a square, 


ae 


Fig. 61. 


area varies with the square of its linear measurements. If 
you make the radius of one circle twice as great as the radius 
of the other, the first area will be four times as great as the 
first. If you make the areas proportionate, the radii must be 
in the relation of 1 to the square root of 2. Both circle and 
square require the more or less tedious computation of square 


80 CHARTS AND GRAPHS 


roots and repay this labor with inaccurate and ambiguous 


results. 
eee 


Fig. 62. 


We can carry this example further, into three-dimensions. 
That is to say, suppose we attempt to show a one-dimension 
fact by a three-dimension chart. Suppose some bright young 
illustrator suggests that, since it is population we are showing, 
we use the picture of a human being for a chart. Now let us 
make the height of one person just twice the height of the 
other. What has happened to the volume cr weight of the 


two persons—assuming that they are similarly shaped? 
Clearly one is eight times as large or as heavy as the other, for 
the volume of cubes or solid bodies varies with the cube of linear 
dimensions. Our pictures give a grossly exaggerated impres- 
sion of the comparison of the two cities. 

To counteract this, we can make the volume or weights of 
the two bodies represented in the proportion of one to two. 


DIMENSIONS AND VARIABLES 81 


But in this case their heights and other linear dimensions are 
going to be in the ratio of one to the cube root of two. You 
will find if you look it up, that the cube root of two is 1.26. 
And you will be even more disappointed with your resulting 
volumes when you have drawn their heights to scale, than 
you were with the resulting surface areas. Meanwhile your 
poor reader will be trying to choose between three ways of 
judging from your picture—height, surface area, and cubic 
volume—with a two to one chance of making the wrong choice. 

The illustration of population by areas or solids is a fre- 
quent type of faulty chart. We have all seen pictures pur- 
porting to show the sizes of various armies, for instance, by 
pictures of soldiers drawn to different sizes. Even the United 
States Census,? has turned out an entire volume of several 
hundred pages, containing many circles illustrating relative 
sizes. It is perhaps the commonest form of deceptive or 
ambiguous chart used—most frequently occurring when the 
author has tried to combine a picture of his items with a chart 
of their mathematical ratios. 

When you see such a chart in the future, learn to ask your- 
self what part of this chart shows the true ratio, height, area, 
or indicated volume. In nine out of ten cases you will find, 
by looking at the data, that the height or linear dimensions 
are in true proportion. You will then realize that the area or 
indicated volume is grossly deceptive and you will not be 


4 [_] 
AJ 
7S 


Fig. 65. The Effects of Comparison by Linear, Square, 
or Cubic Measures. 


2 One of the greatest official accumulations of charts is the Statistical Atlas of the 
Twelfth Census of the United States, in which circular areas were used very exten- 
sively with correct interpretation of values by areas. 


82 CHARTS AND GRAPHS 


misled by it. In any case you will quickly see that the man 
who made the chart either did not know his business or if he 
did, was guilty of shameless intent to deceive. 

In short, the rule that no more dimensions or axes should 
be used in the chart than the data calls for, is fundamental. 
Violate this rule and you bring down upon your head a host of 
penalties. In the first place, you complicate your computing 
processes, or else achieve a grossly deceptive chart. If your 
chart becomes deceptive, it has defeated its purpose, which 
was to represent accurately. Unless, of course, you intended 
to deceive, in which case we are through with you and leave 
you to Mark Twain’s mercies. If you make your chart accu- 
rate, at the cost of considerable square or cube root calculating, 
you still have no hope, for the chart is not clear; your reader 
is more than likely to misunderstand it. Confusion, inaccuracy 
and deception always lie in wait for you down the path depart- 
ing from the principle we have discussed—and one of them is 
sure to catch you. 


CHAPTER IX 
HUNDRED-PER-CENT BARS 


The division of a “whole” into its “parts” is logically one 
of the first steps in any analysis. Usually the graph illus- 
trating this division belongs at the beginning of a statistical 
report. Thus, if your report covers the sales of the company, 
your first chart would break up total sales into the individual 
sales for each line or for each district. The remainder of the 
report, treating of details of the various “‘parts’’ (e.g., lines or 
districts) will then follow a summary chart which has estab- 
lished their relative importance. 

A quantity can always be illustrated by a straight line, or, 
as it is commonly called, a “‘bar.’”’ Bars are the simplest and 
often the best form of erate The total length of the line 
then represents the total value of the quantity. When we speak 
of a line in charting, we do not mean an imaginary straight line 
having neither width nor depth, for that would be invisible 
and could not, of course, be actually used in illustrations. In 
its place we use the bar, with a visible width (and the actual 
depth or thickness of a layer of ink). But it is still proper to 
speak of this bar as being a line or one-dimension chart, for its 
width and thickness are constants, necessary to give visibility 
to the line, and its length alone is significant. 

Now a single bar, illustrating a single figure, will have no 
particular meaning for the reader, because he has nothing to 


Total Foreign Trade 
_ $13,347,340, 777 7 ea ee 


Exports a ~ = Imports = 
$8,108, 986,663 $5,238,352 ,114 


FOREIGN TRALE OF THE UNITED STATES 


Fig, 66, A Simple 100° Bar. 
83 


84 CHARTS AND GRAPHS 


compare it with. There is therefore but one case in which we 
have any use for the single bar, shown by itself. This is the 
case in which we wish to show the parts into which a total 
may be broken up, or of which it may be composed. And 
whatever the quantity or total be which is thus composed of 
two or more parts, we may call it 100 per cent of itself and the 
parts will then be certain percentages thereof. Hence, this 
single bar shown by itself, has come to be called the “100% 
bar,” regardless of whether the total and part quantities be 
quoted in the data and chart in actual values or in relative 
percentages. Irrespective of scale or calibrations, the bar rep- 
resents a whole or 100 per cent, and its divisions or parts rep- 
resent parts or percentages thereof. 


PERIODICALS IN THE UNITED STATES 


(Source:- N. W. Ayer & Son) 


Fig. 67. Many Segments, No Shading. 


In the 100% bar we have the mathematical equivalent of 
the classification chart. Turn back to Chapter III if you 
have forgotten what a classification chart is—or better still, 
don’t turn back, but stop and try to remember it. The classi- 
fication chart sets forth the ideological or schematic relation 
of things. Commonly it displays the parts of which a whole 
is composed. When it does this, it is very similar to the 100% 
bar. The difference between the two lies in the fact that the 
classification chart shows what the parts are but does not tell 
their relative importance or size, while the 100% bar shows 
their relative importance or size, upon the basis of certain 
numerical data. 

Thus the best labelling (for labelling is half the work of 
chart-making) for the 100% bar would seem to be a classifica- 
tion chart placed, obviously, above the bar. As a matter of 
fact, wherever it 1s typographically possible, this is correct. 
Use a classification chart to label a 100% bar, or a 100% bar 
to illustrate a classification chart, and you have the best pos- 
sible combination. Of course where some parts are relatively 
very small, typographical difficulties arise. If you cannot over- 
come these by setting certain words on edge, you will be 


HUNDRED-PER-CENT BARS 85 


Total Casualties 
302,612 


Deaths Wounded 
17,121 221,089 


Killed 
in 
Action 
34,218 


Degree 
Unknown 
46,480 


Wounded 
Slightly 
91,189 


Wounded 
Severely 
83,390 


30 


AMERICAN CASUALTIES IN THB WORLD WAR 
1917-1918 


Fig. 68. Classification-chart and 100% Bar. 


obliged to content yourself with a general grouping of minor 
parts into a single part labelled “Other,” “‘Minor,” ‘‘Miscella- 
neous” or some such rag-bag title for stray odds and ends. 
A further detail of the 100% bar and its labelling, is the 
scale. This should generally be in hundredths or percents. 
The data may be entirely in absolute quantities, but neverthe- 
less the scale should show percentages. To prevent the con- 
fusion of scale and divisions of the bar, the scale should be 
outside the bar, and the best practice seems to be to indicate 
the scale by little notches or short perpendicular lines dropped 
below the bar, from its lower edge. The scale should have 
ten, twenty, or a hundred of these little lines, each indicating 
a division of ten, five, or one per cent. The purpose is to enable 
the reader, by counting notches on the scale, to compare parts 
of the bar with greater accuracy. For the same reason, the 
actual percentage to which each part is equivalent should be 
written or printed below the bar under the center of each part. 
The bar itself, as has been said, should be of appreciable 
thickness. Too light or narrow a bar, such as a thin line, has 
no emphasis or force. But too wide or heavy a bar introduces 
two-dimensional rather than linear conceptions in the mind 
of the reader and sometimes produces undesirable optical illu- 
sions. The width of a bar should be such as to make it clearly 
visible at the distance from which it is to be viewed. The best 


86 CHARTS. AND VGkRaens 


form of bar is generally between a tenth and a twentieth as 
thick as it is long. 

The bar should be hollow, that is, outlined. The segments 
or parts of the bar may also be hallows: but it is better to shade 
them with distinct colors or shadings. Small dots, various 
hatchings (cross-lines), and double-hatchings (criss-cross lines), 
can be used to distinguish the various parts without using 
colors. But where colors can be used, they are sometimes de- 


Clothin, 


(a ___ 


Housin, Fel F&F 


Miscellaneous 


THE FAMILY BUDGET 
Divided as to Classes of Commodities 
United States 
1913 
(Source:- Monthly Labor Review) 
(Note: S&L, Fuel and Lighting; F&F, Furniture and Furnishings) 


Fig. 69. Distinct Shading. 


sirable, for the reason that solid shades are more forceful than 
black and white shadings. Care must be taken in either case, 
however, to see that the various parts are similarly emphasized 
by the color or shade, for otherwise one part will appear more 
important or larger than it really is. A solid black area will 
appear to be larger than a solid white one outlined in black, 
though really of exactly the same size, for the black is in itself 
so much more powerful than the white, and has further gained 
by absorbing its black outlines. Experiment will soon show 


Exporte 
8,108,988 ,000 


| 


Atlantio Coast Gulf Const 
= 1445 386 


Imports 
| $5,258,362 ,000 


Atlantic Coast 
3,763 ,649- 


FOREION TRALE GATEWAYS 
Porte of Exports and Imports by Specified Groupe 
Onited Btates 
1920 


Value 


Fig. 70. Two Bars are Easily Compared. 


HUNDRED-PER-CENT BARS si) 


whether an optical illusion !is being introduced by the shadings, 
and those combinations which will bring out the various parts 
equally. 

The data for the 100% bar need be no more than a list of 
the parts of which the whole is composed, with their respective 
percentages, and either with or without their respective abso- 
lute quantities, according to your wish. If the quantities are 
important, or you think that someone is likely to call for them, 
add them and forestall criticisms; if they ‘are unimportant, 
they can be safely omitted. While the percentages are almost 
always desirable and are best placed below the bar, as part of 
the scale, the absolute figures or data, if inserted, are best 
placed immediately over the bar, as part of the classification 
chart which is used for the labelling. 

That data of this character calls only for a single dimension 


Year 1913 


Food 


Lec 1920 
Clothing 


THE FAMILY BUDGET 
Divided as to Classes of Commodities 
United States 
1913, Dec 1920, Dec 1921 
(Source:- Monthly Labor Review) 
(Note: Fal, Fuel and Lighting; F&F, Furniture and Furnishings) 


Fig. 71. Comparison of Three Different Years. 


chart, according to the rule of chart dimensions is obvious. 
For the individual figures, with their stubs, in the table of 
data, can be shifted freely up or down the column or across in 
the row in which they are tabulated and the stubs therefore 
cannot be said to form a variable. In another chapter we shall 
consider another way of presenting the same data, in which 
each figure forms a separate bar, and the series of figures in 


1 For a discussion of the various optical illusions to be avoided, see Willard 
C. Brinton, Graphic Methods for Presenting Facts, Engineering Magazine Co., pp. 358, 
359. See also Appendix D. 


again oan pee AG bE c 
one gals aie a, a ae of the 104 ) 


shown on the 100% I bie F or all te series oe pine! bar 
charts discussed in a later chapter must be used. 


CHAPTER X 


PIE-CHARTS 


Throughout your study of charts you will find some which 

are more useful for popular consumption than others, but you 
will not find many which are more purely popular in appeal 
than the 100% circle or pie diagram. For analytical purposes 
it has nothing to recommend it, but for sensational values it 
is in general without an equal. If you are research-bent, you 
may safely pass by this chapter on popular exegesis, but if 
your object is advertising, you will seize it to your heart. 
i; We have just seen how a single bar can be taken to repre- 
sent 100% and can be cut up into segments or parts the lengths 
of which correspond to the relative sizes or percentages of the 
various parts of the 100%. ‘The fact that the line is a unit, 
and so long as it remains the whole, can never be more than a 
unit or 100%, should suggest something. It should suggest 
that the total length of the whole line is relatively unimportant. 
It is unimportant because the reader is not asked to compare 
the total length of the line with the total length of any other 
line. There is no other line to compare it with, unless a second 
100% bar is lying around handy, in which case the second 
would presumably have the same length, because it too repre- 
sents 100%. Hence, you will say, why have any total length 
at all? 

Centuries ago it was a moot question among philosophers, 
whether the Lord could make a yardstick which was endless. 
Then someone suggested that the yardstick be bent into cir- 
cular form and the question was dropped. Let us perform the 
same operation on a 100% bar. Imagine, if you wish, that 
the bar is so very thick for its length that while one ae be- 
comes the circumference of the circle, the other shortens down 
to and becomes the center of the circle. Division lines between 
the component parts of the bar become rays or radii of the 
circle and serve to mark off the corresponding component seg- 


89 


~ 


gO CHARTS AND GRAPHS 


ments of the area of the circle. Here you have in a nut-shell 
the pedigree of the pie-chart. 


IMPORTS INTO RUSSIA 
1921 
(Source:- Russian Information and Review, London) 
(Grand Total, $124,281,000) 


Other countries 


Norway and Sweden 


United States 


Germany 


Fig. 72. A Simple 100% Circle. 


It is now time to let you into the secret that the rule of 
dimensions of charts, which you doubtless memorized in a 
previous chapter, has apparent exceptions. The pie-chart is 
one of them. For few readers will judge quantities by either 
the arc at the perimeter of the circle or the subtended angles 
at its center—on the contrary most of them will judge entirely 
by the areas of the segments. In short, the pie-chart appears 
to be a two-dimension (area) chart used for one-dimension 
data. The fact is, however, that, as in the case of the 100% 
bar, the area of the chart varies directly with one dimension, 
the other dimension being constant. In the 100% bar the 
width of the bar was constant in the 100% circle the radius 
must be constant for all circles compared. Then the area” of 
the segments varies directly with their arcs or angles and the 
chart has but one significant dimension. It is only an ap= 
parent exception to the rule. 


PIE-CHARTS ¥ 


The disadvantages of the pie-chart are many. It is worth- 
less for study and research purposes. In the first place, the 
human eye cannot easily compare as to length the various arcs 
about the circle, lying as they do in different directions. In 
the second place, the human eye is not naturally skilled at 
comparing angles—those angles at the center of the circle, 
formed by the various rays or radii and subtending the various 
arcs. In the third place, the human eye is not an expert judge 
of comparative sizes of areas, especially those as irregular as 
the segments of parts of the circle. There is no way by which 
the parts of this round unit can be compared so accurately and 
quickly as the parts of a straight line or bar. Moreover, when, 
as frequently happens, several pie-charts are shown together, 
the various slices in one chart cannot be so easily compared 


LAND AND SBA ARBAS AREA OF CONTINENTS GSOGRAPHY OF THE LAND OF THE SARTH 
Total surface Total land ares Total land ares 
396,940,000 square pilee 67,255,000 square miles 57,266,000 equare wiles 
2p, 
X : 
RACES OF MAN LABOUAOE POPULATIONS 
Total population Totrl Lda 


3,702 ,000,000 


Bastorn 
Catholic 


Worth 
3 


an. 
South 
America 


$ LECTS 
RELIGION POPULAT LOWS CONTINENT POPULATIONS DISTHIBUTION oy OR DIALECTS 
Total population Total population alese 
1,792 ,000,, 1,702,000, . 


\ 


Fig. 73. Accurate Comparisons Cannot be Made. 


92 CHARTS AND GRAPHS 


with the corresponding slices in the next, as can the various 
parts of one 100% bar with corresponding parts of another 
bar. The two bars can be placed one above the other, so that 
comparison from one to the other can be made at once, but 
no arrangement of the two circles will make comparison so 
simple. 

In the labelling of the pie-chart, you will furthermore en- 
counter typographical difficulties. It is not ordinarily a good 
thing to make a reader crane his neck at various angles to read 
writing along every point of the compass, so you should not, 
as so many do, write on radii from the center of the circle. 
On the other hand, unless the chart and its segments are very 
large as compared with the size of the printing, you will intro- 
duce tricky optical illusions if you write all labels in the same 
directions inside the segments. 


99 
cents 


1914 1915 


1918 


PURCHASING POWER OF THE DOLLAR 
OF 1913 


when used for food at retail 


U. 8. 
Fig. 74. The Less Detail, the Better. 


PIE-CHARTS BF 93 


Sometimes, the best rule is to put the labelling away in a 
key or explanatory list of the shades or colors of the various 
segments. Only if the labels are very brief, and your segments 
are all large, can you stow the labels into the segments without 
greatly altering their apparent sizes. When neither plan is 
feasible, and you feel that you must have each segment, how- 
ever small, immediately labelled, place the labels outside the 
circle, adjacent to the proper segments, with the printing in as 
nearly the same general direction for all as you can arrange, 
so that the apparent sizes of the segments are not confused 
by printing and the reader need not climb around the edge of 
your chart to decipher it. 

As a general thing, however, there is one part of the label- 
ling which can always be attached to the chart, namely the 
figures of the percentages for each segment, which i> the bar- 


Retait Foop EstaBLisHMENTS 
New York City 


1921 
(Totrat Numser, 75,412) 


9% 


PusHtarts 
i 


WAGONS 


CONFECTIONERS 


BAKERIES 


BUTCHER. 
GROCERY 


STORES 


Fig. 75. Labelling is Difficult. 


chart were placed immediately below the bar. These figures 
should always be placed close to the segments, but usually out- 
side the circle, so that the reader who wishes to have the precise 
percentage figure represented by a segment, can always do so. 

The scale (without scale-figures) may be placed inside the 
‘circle and unlike the 100% bar may or may not show in the 
finished chart. Special paper is marketed for these charts with 


“Laphug 409 “4 fo 
“0155142 q— pauinsse St Jo]I@101 03 J9INIDeynUeUL Woy aes IaN [lela qe 4 
Sulj[as pue 3ulyiopd Jo 9p1are ay2 Jo ainqovjnuew {foos 29 u0209 ‘Toyrea] MvI ayy 
WOIJ YI0]9 JO JaYyIea] Butmedaid :sassao01d 9eIYy} YFnoIYI parsed ‘sayio]d Jo tins 


= S$ © pue “Aaiys 7g & ‘saoys yo ned S$ & Je J9uMsUOd ay 03 Is09 ay jo siskjeuy 

= "SHIYD AL]JOq WweT]e0xq -g/ “314 

& yoav7 

) ~h 

Q g 4, UBTIWL3Y %, 

2g ae 

ok 2 
= ee Ze m 

ce Ss --“yaunLovsnNVnd |S, 

3 3 

ah Do 

x 

CO 


TWIYaLWW 
MVY 


TWIYSLVW 
IMVa 


LINS Ses AYIHS Ze SA0HS OF 


94 


PIE-CHARTS 95 


GROCERY STORES 
18.91 % 


Bad debts 1.94 
Depreciation2.14 
Ins. 1.57 
Adv, 3.72 
STORES Gen. 1.10 
ce 


Delivery 2.53 
Adv. 1,83 
Rent 3,07 
Sal, 8.46 


JEWELRY STORES 
26,81 % 


ZT 


7777) 


HARDWARE STORES 
20.41 % 


Adv. 1,12 
Gen. 2.01 
Rent 3.41 

Sel, 10,11 


Delivery 1,02 
Depreciation 1.12 
Ins. 1.08 
Adv. 4.67 
Gen. 4.16 


Rent 3.24 
Sal. 9.66 


MEN'S CLOTHING STORES 
23,27 % 


Depreciation 2,16 


SHOE STORES 
24.22 % 


Heat end light 1.108 
Ins. 1.03 
Adv, 
Gen. 
Rent . 
Sal. 10. 


Salaries 
Rent 
(UM) General Expenses 


GRR Adverticing ‘ 


(-__J Insurance and Taxes 


Fiieeieg Deprecieticn 
Delivery 
Bad Lebts 
Heat and Light 
Miscelleneous REILE 2 STORREESTENOES 


Percentage of Expenses to Sales 
with chief items of expense 


(Seurces:* Horverd Bureeu of Business Kesearch 
ard various reteil research orgenizations) 


Fig. 77. Shading the Segments to Increase Popularity. 


the circle printed in and already divided into a hundred parts 
by small notches within the circumference. The use of this 
paper will save you much time if you wish to make the seg- 
ments accurate in size. It is tiresome to use a protractcr 
marked off into 360 degrees, and to calculate the decimal 
equivalent or percentage in degrees. Unfortunately, the metric 
system has not been applied to circular measurement so as to 
give any circular or angular measurement which employs a 
decimal system. 


96 CHARTS~ AND GRAPHS 


The advantage of the pie-chart is psychological. It in- 
stantly commands the reader’s attention. A circle is, of all 
geometrical patterns, the easiest resting spot for the eye. The 
fact 1s well known to advertisers, who frequently use circles 
and circular outlines to draw attentica to their advertisements. 
Hence if your chart is designed for publication, or for presenta- 


af 
4 


Fig. 78. How to Calibrate the Circle. 


Bisect OD at E and project EF equal to EA. Project the chord AG equal to 
AF. With dividers set for this distance, lay off from A the points, G, H, K, and 
J (as in upper right-hand diagram). Similarly lay off four points from B, C, and 
D (as in lower left-hand diagram). Erase all other marks and calibrate the 
twenty points so found, to form the finished circle (as in lower right-hand diagram). 


tion to readers whose attention may be easily diverted, you 
will find the pie-chart a powerful means for presenting your 
facts. Attention will be focused upon it at once, and it is as 


Ny 


PIE-CHARTS 97 


simple to understand as its name—far too simple for anyone to 
misunderstand. Because it is circular, there is no question 
but that it represents a whole and the various slices of the pie 
belong to their respective items. 
Cost or THE Wortp War To THE Unrrep States 
EsTIMATE ON Jury, 1621 


Granp Tota, Cost—$50,168,625,707.16 
(Source: Woritp ALMANAC) 


% ivasTmensy 
. ere 
LOANS Te ALLIES 
10 faci (ove veneer) ‘ 
ENSIONS 
JuUCcime FROM LOE Ou None 
crv WAR 


MILITARY cosr 
wf o4,0. 0,000,000 


Gesecay. novsros) 


+o 
Gev'r Loans 


43 


Fig. 79. Many Segments, No Shading. 


A very sound use of the pie-chart occurs in the case of 
financial data. Here the whole circle or pie can be spoken of 
as a “dollar,” and the various segments, the parts of the dollar, 
so many cents (or percentages) each. Charts of this type have 
been used to show the distribution of costs in a plant, or the 
parts of a financial budget, or the shrinkage of the dollar in 
high-cost-of-living studies. Through the fortunate coincidence 
that our metal currency is round and our dollar divided into a 
hundred cents, the shape and the labelling of the chart both 
find immediate understanding in the mind of the reader, 


FT HIS SHOWS 5 
WHATEBECOMESS 
EAVERAGERN 


Fig. 80. A Pie-chart in Metal. 
The “Swift Dollar,” as it was called, on one side of which a chart shows the 
division of income from sales. 


The pie-chart must therefore be accepted as an advertising 
medium of value. It has strong popular elements. But it has 


98 CHARTS AND GRAPHS 


no place in the statistical workshop, or research laboratory. 
Before using it in the place of the simpler and sounder 100% 
bar, you should carefully gauge your audience or readers, and 
only if you believe that you have begun to strain their interest 
should you judiciously insert the pie-chart. In a sense, it 
might be construed as an insult to a man’s intelligence to show 
him a pie-chart, but the insult is not often resented. For if 
your main object be to get a story across, you are justified in 
taking that means which will encounter least resistance, and 
in making your story as simple as possible. For publicity 
purposes, the pie-chart is therefore almost invariably better 
than the bar. 


ee 


CHAPTER XI 
BAR-CHARTS 


A series ot quantities or values can be most simply and 
often best shown by a series of corresponding lines or bars. 
All bars being drawn against one and the same scale, their 
lengths vary with the amounts which they represent. In a 
previous chapter, the 100% bar was described, in which a 
single bar, whose total length had no significance, was divided 
into parts in order that the relative size of the parts might be 
seen. In this chapter we propose to use several bars, which 
are not divided up into parts, but which can be compared as to 
their total lengths, in order that their relative sizes may be 
seen. While this new method could be employed with the 
same data, it is generally more useful for data in which the 
various items are not being shown as parts of a total, but as 
individual and co-equal totals in themselves, 

BUSINESS FAILURES 
Smount of Liabilities t 
United States 


1920 
(Source:- United States Census) 


(Millions of Dollars) 
© 10 20 30 40 50 60 70 80 90 100 110 120 130 
Manufacturirg 127,992,471 
Trading 88,558,347 
fgents and Brokers — 78,570,987 


Banke $0,708,300 


Fig. 81. A Simple Bar-chart. 


Bar-charts are most flexible and can be varied to suit the 
individual whims of the maker. In general, however, there is 
one style or form which will be found most satisfactory. It 
consists of a horizontal grouping of bars alongside of the data. 
The chart is arranged in tabular form, with items or stubs in 


99 


100 CHARTS AND GRAPHS 


a column to the left, with figures in a column beside the stubs 
and with bars in a column beside the figures. Several columns 
of figures are sometimes desirable, just as in the table of data, 
to show sources or original figures from which the charted 

Area Pop. 


Population . per 
square miles #q.mi. © 10 20 40 60 80 100 120 


(Scale of Population per Sq. Mile) 


TOTAL 1,702,520,000 57,255,000 29.6 
Europe 464,681,000 3,873,000 120.0 
Asia 872,522,000 17,206,000 50.7 


North America 150,000,000 8,589,000 16.3 


Africa 142,750,000 11,623,000 12.3 
South America 56,340,000 7,570,000 7.4 
Australasia 16,230,000 3,313,000 4.9 
Polar Regions oo: 6,082,000 eee 


DENSITY OF POPULATION OF THE EARTH 
by continents 


Fig. 82. Detailed Data may be Included. 


figures are obtained. In any case, the bars should represent 
the most important set or column of figures, and there should 
be normally but one column of bars. 

There should be but one column of bars because the bars 
can be advantageously compared only when they are side by 
side, one below the other. They should all begin at a uniform 
point or distance from the figures, so that their lengths can be 
compared out at the far ends. They should be the last column 
on the page, because their uneven lengths make further col- 
umns wasteful of space, and the addition of further columns 
introduces optical illusions which should be avoided. If a 
column of bars were to be followed by a column of figures, 
the reader would be apt not to compare the lengths of the 
bars, but their shortage from the last column. 

It is a very common error to place the data inside the bars, 
for by so doing the reader is led to compare those parts of the 
bars which are clear of figures, rather than the entire lengths 
of the bars. This optical illusion exaggerates the difference 
in lengths of bars. Another mistake which is often made is to 
place the data out at the varying ends of the bars, for here the 
reader is led to compare the lengths of bars plus data, rather 
than of bars alone. Here the optical illusion minimizes the 


————— ee es 


BAR-CHARTS toy 


difference between bars. The proper place for data is in a 
column at the left of and immediately before the bars them- 
selves, with no more reading matter to the right of the bars. 

The scale for a bar-chart should be placed at the top of 
the chart, immediately above the uppermost bar. A field in 
fainter color (green is most useful for chart-fields) or thin lines 
should be drawn into the chart by extending down from the 
scale a few lines which mark off convenient distances on the 
scale. Thus the reader is enabled to compare lengths of bars 
far distant, by noting their relative positions against the field 
or background. Care should be taken that the lines of the field 
do not cross the bars, else the field will cease to be a back- 
ground and will become a screen in the foreground. 


(Dollars) 


(Scale of Hundreds of thousands of dollars) 
Republicans: i Cs) 2 


2 


Ger. Leonard Wood 1,773,303 Ea 
Gov. Frank 0. Lowden 414,984 aoe 

Sen. Hiram W. Johnson 194,393 
Herbert Hoover 173 ,542 
Sen. Warren G. Harding 113,109 


Sen. Miles Poindexter 77,150 8 


Gov. Calvin Coolidge 68,375 F 
Dr. Nicholas Murray Butler 40,550 
Sen. Howard Sutherland 4,145 
8en. Joseph Irwin France 1) 
Democrats: 

A. Mitchell Palmer 59,610 
James H. Cox 22,000 
James W, Gerard 14,040 
Gov. Edward I. Edwards 12,900 


Sen. Robert L. Owen 6,595 


Sen. Gilbert M. Hitchcock 3,357 


William G. McAdoo 0 


PRESIDENTIAL CAMPAIGN BXPENDITURES 
United States 
1920 
(Source:- Report of Senatorial Committee) 


Fig. 83. A Long Bar Broken to Save Space. 


The bars should be of uniform thickness or width, as it is 
the variation in their length which is significant and variations 
in width would produce area-illusions. This rule is obvious— 
so obvious that where a large number of bars are shown and 
the reader is already thinking in terms only of the lengths of 


CHARTS AND GRAPHS 


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BAR-CHARTS a 


The extra width given this bar need be only slight to make it 
stand out from the rest as a sort of type or normal against 
which the individual bars can be measured. And this should 
be done only when there is no danger of the reader’s attaching 
importance to areas. . 

It sometimes happens that you desire to show two tables 
on the same subject, one giving a large number of individual 
items and the other a few sub-totals. It is a good plan in 
such cases to place the two on separate sheets that face each 
other, so that they both show at the same time, the one giving 
summaries of the other in the form of sub-totals. Where the 
summaries are not averages, but are true summations or 

TRADE UNIONS OF TH WORLD 
Estimated Membership of trade unions in twenty ohief countries 


1919 
(Souroe:+ International Labor Office) 


TOTAL 32,680,000 Sets ii = TT Ripa $= 
Australia 628,000 [_] ; 
Austria 772,000 me 
Belgiun 760,000 C_] 
Canada 378,000 [J 
Ceechoslovakia 657,000 [_ ] 
Denmark 360,000 [_] 
Finland 41,000 
France 2,600,000 
Germany 9,000,000 
Great Britain 8,024,000 
Hungary 600,000 
Italy 1,800,000 
Netherlands 626,000 

New Zealand 100,000 
Norway 144,000 
Roumania unknown 
Serbia (Jugoslavia, 20,000 

Spain ; 211,000 
Sweden 339,000 
Switzerland 224,000 
United States §,607,000 


Fig. 85. An Alphabetic Arrangement, 


104 ‘CHARTS. AND GRAPHS 


totals of the items they include, you will find it well-nigh im- 
possible to draw both charts to the same scale, without having 
one or the other so large or so small as to be useless. You will 
therefore be obliged to shorten your scale, that is, use a smaller 
scale for the summary-bars than for the individual bars. 


In such cases it is not a bad plan to make the sub-total 
bars wider. In exactly the same proportions as you condense 
the scale, you should thicken the bars plotted thereon, so that 
the areas of the summary-bars will be equal to the combined 
areas of the individual bars of which it is the total. This is 
one case in which a slight use is made of the conception of 
area, or two dimensions, but it is negligible, for the reader is 
not called upon to compare areas—only lengths—in all cases, 
and the thicker bars remind him that the scale has been short- 
ened on the sub-total chart and prevent his making confusing 
comparisons between the two charts. In short, when the group 
item in a series is an average of individual items in the series, 
it can be shown on the same scale, but where it is a total or sum 
of individual items, it can not be shown on the same scale 
without making the individual items small, but can be shown 
on a separate chart in reduced scale and with correspondingly 
increased thickness. 


(Seale of Dollars) 
fo) 20 40 60 80 


1800 15.63 
1810 7,34 
1820 9.44 
1830 3.77 
1840 e21 
1860 2.74 
1860 1.91 
1870 60.46 
1880 38.27 
1890 14,15 
1900 14,55 
1910 11,34 
1920 228.63 


PERCAPITA PUBLIC DEBT 
Less Cash in Treasury, July lst 
United States 
1800-1920 


Fig. 86. Historical Data Must be in Order. 


BAR-CHARTS 105 


The arrangements of items in a bar-chart should be a 
matter for careful study, and no arrangement should be chosen 
which is not the best adapted for the special purposes of the 
chart. It is impossible to lay down absolute rules, for each 
case varies, and the author of the chart must rely on his own 
judgment rather than on rules of thumb. But in general the 
principles explained in a previous chapter on work sheets 
apply here, as the bar-chart is closely akin to the mere statis- 
tical table. When the items are historical, the earliest dates 
should be at the top and the last at the bottom of the chart, 

THE CAUSES OF FIRES 
United States 


1915-1919 
(Source:- Nat'l Board of Fire Underwtiters, N. Y.) 


(Value) SOALE OF MILLIONS OF DOLLARS 


inet) mes O0. 80-0 406 8G. Leo a2 
TOTAL 1,138,000,000 a 0 780 90 


Electricity 84,100,000 & 
Matches and smoking 73,500,000 
Defective chimneys 66,700,000 fi 


Stoves, boilers & pipea 65,100,000 


Bpontanscus combustion 49,700,000 
Lightning 39,800,000 
Sparks from machinery 31,900,000 
Sparks on roofs 29,300,000 
Petroleum & produots 25,900,000 


Sparks fram combustion 25,100,000 

Inoendiarism 21,600,000 esc naa 
PATTER 
as 
peck seats] 
—_— 
'] 
B 


Open lights 14,000,000 


Ashes,ooals & open fires 11,810,000 
Gas, natural & artitic. 10,200,000 
Explosions 10,160,000 
Hot grease, oil, tar,eto 4,490,000 


xubbish and litter 3,510,000 


Steam & hot water pipes 1,850,000 


Fireworks, firecrackers 1,600,000 


Fig. 87. Placing the Most Important First. 


the order being strictly chronological. When the items are 
geographical, as for example a list of the States in the United 


106 CHARTS AND GRAPHS 


(Scale ef Millions of Members 
O2Y (2784 6 6re7 628 30 5 16 17 16 
Pot Pays pes] fe fos oe IE lets est enue bey ad 
oh] | Fa | 
Rmnha 
ee 
fae as [eras 


Soman Catholte 17,549,324 


v. Be (white) 6,328,476 


Faptiet (white) 4,389,769 


Paptiet & M. EB. (colored) 4,191,257 


Lutheran 2,451,997 
Presbyterian 1,603,033 
Disoiples of Christ 1,193 428 
Protestant Bpiscopal 1,065 ,825 
Congregational 608 ,122 
Mormons 494,388 

RELIGIOUS DENOMINATIONS OF UNITED STATES 

Betimated Membership 


1919 
= Source:- “Year Book of Churches” 


Fig. 88. The Arrangement in Order of Size is Popular. 


States, a geographical arrangement (either in the order given 
in the Census volumes or in some specially designed order) is 


Mines of Central Competitive Field = NNT 


NEARLY 40,000,000 TONS PRODUC 
WER: 


ARAB DUE TOALL TRANSPORT 

[ra 5 metensae| 

el cee 

CL asia 

1 ee ees 

[8 | RET a oe | 

1 ae ae. 

C1 ces ae a 

re ene 

eee | 

C2 | | 6 | 6 | ELSTON To 

Pe Tye i 

Cr ister ill 

i a na =m HA WAL RS 

CeCe Riso ray ae oe 

Aver Ohio mea ew m fs oe res 

A bihurinois ae Ba a ielihpeceoe eee 
DLENESS_C 


Fig. 89. The Gantt Idleness Chart. 
Showing the failure to produce to capacity, together with data analysing the 
failure. This chart is one of a series of such charts on the coal mining industry, 
appearing in The Dial—Reproduced from “The Life and Work of Henry L. 
Gantt,” American Society of Mechanical Engineers, paper of Mr. Polakov, 


BAR-CHARTS 107 


preferable. A mere alphabetical arrangement of the States 
would have little to recommend it, as the reader of thechart 
must be presumed to have sufficient intelligence not to require 
a dictionary of the country. The popular arrangement is in 
the order of magnitude of the data presented, so that the 
longest bars are at the top, but unless the reader’s sole purpose 
is to glimpse the names of the leading States, this arrange- 
ment is useless, as it lacks comparability with other charts. 
In general the arrangement of the items should be such as to 
afford the greatest aid to such analysis and comparative study 
as the chart may be subjected to. 


{ Rigiens Ss] 812,728 
Soot land 254,567 
(aie : 
{_Irelend —_—*’:«1, 037,233 
[_torvey ) 388,062 
[Sweden | 85,680 
rs ere 
191,766 
62,600 
ee 
116,489 
[_Prence 118, 509 
34,921 
[ Germany 3,884, 102: ! 
[Poland ____—‘|-1,130,070 
(_sungery 
2 D 
if 
: 3 
. 
aie 
fy? 
i 


Han) 


j Other 47,729 


Fig. 90. Classification-chart and Bar-chart, 


108 CHARTS AND GRAPHS 


The bar-chart should be provided with column-headings 
for the data just as the work-sheet, but even more carefully 
prepared. Small keys or samples of the bar-chart can be in- 


ae 

: 14 UVING 967.1921 EARNINGS 
MENS CLOTHING 271% 
BOOTS- SHOES 253% 
WOOLEN 230% 
CAR BUILDING 2309% 
SILK 228% 
CIGARS 222 Ye 
ALL TEXTILES 
PAPER 


HOSIERY-UNDERWEAR 
COT TON FINISHING 
COTTON MANUFG 
LEATHER 
AUTOMOBILES 
IRON- STEEL 


TOTAL 


Fig. 91. 


Average weekly earnings of workers in representative establishments in different 
industries throughout the United States, and the cost of living. (1914=100%). 
—Permission of Mr. Carl Snyder. 


-————— 


— 


FREQUENCY RATES s Qi ] 
NUMBER OF CASES HR RTO Ray Pe OLA Ye 
gestae SEVERITY RATES 
300 DAV rir DISABILITY a an DISABILITY DISABILITY NOS OF wORK NO Dard 
WORK Hee RMAN|TEMPOR| TOTAL ATH PCRMANTEMPOR, TOTAL || DEATH TOTAL be 300 DAY WORRER ) 
BS cnr ARY ENT aRY 
COLUMN A 8 c D £ F GS H 1 J cal $ oe “5 20 25 


Fig. 92. An Office Record Form to Include Bars. 


—— ee 


BAR-CHARTS meats 


serted under the heading of the data which is plotted to show 
to which figures the bars belong. The scale of the bar-chart, 
above described, forms the heading for the bars themselves. 
When a series of charts are shown together in the same report, 
the charts of absolute figures can be displayed in typing and 
bars of one color, such as black, and the charts of relative per- 
centages, per capitas, or averages can be distinguished by 
typing and bars of another color, such as red. In general a 
judicious use of colors will assist the column-headings and 
titles of the charts in differentiating charts in large sets. 

The technique of bar-charts is so simple and they are so 
very effective, that they should be used freely in printed text- 
matter. No drawing or plates are needed. Printers have 
“rules”? as they call them, which can be used to make solid 
bars, and these rules can easily be set up together with the 
type. The scale and field can be omitted and the bars alone 
will effectively tell the story of the main figures in the table. 
The combined table and chart can be used in printed text just 
as well as the table alone. 


NUMBER OF CASES FREQUENCY RATES SCVERITY RATES 
; = (Phe 1000 S00 DAs woaeees) f (Pre scones woania) EVERITY RATES 
= i= eae di ee a Sév ATE 
pee [CRIS ABH ry” = | py Disapity yg | DISABILITY | (uoss oF won ne Das 
300 Day | DIATH/peamawtempoR | TOTAL |DEATHippemad=empor| TOTAL DEATH Io¢amadtemppny FOTAL S 
WORKERS eT AR ent | 4py ENT | apy meng 
COUMN Al 6 Cc D tae GS A T 5 cape reat G ae ive a 
Total 115,703} 37 | 411 |13,199 [13,647 | 30 {3.60 | 114 i0]11€.00 |] 2.90] 1.60 Pal 60 
Ships 6,615] 3 | 15 | 1,422 | 1,460 | S [2.3 |215.0 }217-8 14.1 ]1 6-/2.31 6.0 H 
ining 3,994 12 755 767 3.0 | 189.0 | 192-0 1.2 [1.9] 3.0 il 
2 . 6.4 9.0 1] 6 |1.6] 4.2 
Cranes 4,362 Tale 48. 813 629 | .2 |3.4 | 186.4 | 19 
Steel 2,692} 1] 16 43a |, 455 | .4 [59 |162 7]169.0 13.3 ]2.9 ]1.5) 7.7 
Engines 31,229] 22 | 160 | 4,348 | 4,530 | -7 1 | 139.2 | 145.0 3}2.9 /1.4 po 6 
Electrical 35,674! 5 | 100 | 3,455 | 3,560 | 1 |2.8 | 96.8] 998 1.3 ]11/ 1.0/3.6 
Porer 2 226 9 186 195 4.0 | 85.0 | 89.6 PASVE | aes Abeer 
5 2 61.9 3 9 ott 8 &) 72.4 
1 Tools 24,359| 3 | 68 | 1.486 | 1,557] .1 /2-8 | 61.9] 639 ray 2.4 BD 
i 
6 96 ‘ 5 5.0] 68.9 | 4.0}11 5.8 
Unclassified 4,552 2 16 296 $4 4a 3. 65 
| 
| | 


Fig. 93. Bars as Part of the Office Record. 


110 CHARTS AND GRAPHS 


United States 3.71 
Mnited Kingdom 1,56 


Belgium 1,02 
Pruesia 2.26 
Austria 1.15 


PATALITY RATE IN COAL MINING 
Fatal Acoidents per 1000 Workers 
Specified Countries, 1919 
(Source: Bureau of Labor Statistics) 


Fig. 94. Typewritten Bars for Typed MS. 


When manuscript is typewritten, the bars can, if desired, 
be Pe in, using such letters as “x” or better still, “‘x,” “‘o”’ 
and ‘‘m”’ printed one over the sche This obviates the need of 
aoe and illustrates excellently any tables which you are 
obliged to insert in the text. It has the further advantage of re- 
producing on carbon', hektograph, or mimeograph copies 
(though the latter can be secured also in drawings by the use 


of a special stylus). 


1Not only can various combinations of colors in typewriter ribbons, be obtained, 
but also various colors in carbon paper, which can be successively inserted in the 
typewriting machine to reproduce the desired effect. Black and red are, however, 
usually sufficient, 


— 


—— 


CHAPTER XII 
COMPOSITE BAR-CHARTS 


What man has done once, he can do again, and since we 
have put several single bars together to make a bar-chart, we 
can put several bar-charts together to make a compound or 
multiple bar-chart. The single bar represented a single figure, 
the simple bar-chart a series or column of figures, and the 
compound or multiple bar-chart will illustrate a series of series 
(or columns) of figures. For the sake of convenience, we can 
divide these last into two classes, one of which we may call 
the compound and the other the multiple bar-chart. Nomen- 
clature is of little importance but a precise use of names will 
help to distinguish two radically different forms. 

Where each bar in a bar-chart is divided into parts, as was 
the single 100% bar, the name compound bar-chart is sug- 
gested. In such cases each bar is really a 100% bar by itself, 


BUSINESS FAILURES 
Amount of Liabilities 
United States 
1916-1920 e 
(Source:- United States Census) 


Manu- ra- Agents, Bank- Grand (Millions of Dollars) 
fac- ding Brokers ing total e “a8 


‘AO sf mm ooo 


1916 13 91 32 10 2OTOLZLLI LLL LULA 


1917 80 70 33 16 201 LLL 
1918 3 58 32 5 7 SEREE ER BREEERS 
3919 62 38 24 17 lS OUZZZZLL LL 2 

1920 128 89 79 61 346 bad ban 


Fig. 95. The Compound Bar-chart. 


but its length may be no longer constant and uniform, and 
may be made to vary in the fashion of the simple bar-chart. 
Its scale and labelling follow the same form. Data again 
should be at the left of the bars. The scale should be above 
the bars with a field projecting it downward behind them. 


IIL 


CHARTS AND GRAPES 


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COMPOSITE BAR-CHARTS ga) 


The bars should be placed beside and on the same line with the 
data which they illustrate. As to the arrangement of the 
columns of data, it is best to have the column of totals, which 
the entire bar represents, at the beginning or end, and then 
beside it, in the order in which the parts will appear in the 
bars, the columns of various parts. Shading of the parts will 
follow the rules given in the chapter on 100% bars. 

We will find two distinct types of the compound bar-chart, 
each belonging to a certain type of data. If the data ex- 
presses the values in absolute quantities, the totals for the 
various items or stubs (lines) will not necessarily be equal—in 
fact will very rarely be equal. So that the chart of this series 
will have bars of unequal total lengths. (Needless to say, the 
widths of bars will always be constant and uniform, save for 
the special exceptions noted in alater chapter.!) Such a chart 
has its scale and field measured off in units of absolute or 


COAL RESERVES (Unmined) or THE WORLD 


1920 Estimates in millions of tons 


Total all Anthra- 


ee sites SCALE OF BILLOONS OF TORS 
a Gl > 500 1,000 1,500 2,000 2,500 3,000 3,500 

GRAND TOTAL 7,460,506 642,105 

United States 3,538,506 16,1635 

Canada 1,361,000 2,000 

China 1,097,000 427,000 

Germany 467,000 A 

Groat. Britain 209,000 12,000 

Siberia 192,000 - 

Australia 183,000 - 

India 87,000 - 

Russia (in Burope) 66,000 41 


Union of 8. Africa 62,000 13,000 


Austria 69,000 - 
Colombia 0,000 - 
Indo-China 22,000 22,000 
France 19,000 4,000 
Belgius 12,000 : 
Spain 10,000 2,000 
Spitzbergen 9,000 - 
Japan 9,000 - 
‘Bolland "6,000 - 
Other Countries 24,000 8,000 


Fig. 97. j Very Small Segments May Be Shown. 


1Cf. Chapters L. and LI. 


114 CHARTS AND GRAPHS ~ 


actual quantities, such as dollars, pounds, tons, or whatever 
the unit quantity be in which the data is expressed. It is an 
‘‘absolute” chart, or chart of absolute values. 

The other type of data is derived from the last, but is 
often more significant. In it the quantities have been turned 
into percentages, in each case of the totals for the line or stub. 
The totals therefore are in every case 100% and the bars are of 
uniform length—literally a series of 100% bars. The scale for 
such a chart is measured off in percentages instead of actual 


PREBoWAR OCCUPATIONS 
of the wage-earning population 
C8 SEVEB EUROPEAN BaTIONS 


Agric Com frans- Mine Manu- Con- fTexrt- Misc’ 

eul- mer- porte- ing, fact- etruce iles, ella- (Percentage) 

ture otal tion eto. uring tion Dross wmeousg 6 10 20 380 40 50 60 

wg SS ws Co ealcalanio alee 
Mmglend 1267 11.6 8908.2 B07 6.6 1.2 32.8 YUL EHO 

ER Ta 2 eee ST) 

Selgius 21.9 11.8 2.0 6.6 6.0 1.3 “4.7 29.8 OP 
Germany 386.) 6.5 2.9 3.5 7.0 7.0 9.1 2.3 
Trame 41.4 6.5 2.9 1.6 4.4 4.2 12.6 26.4 
Ttaly 69.3 3.4 3.1 0.9 2.1 6.0 14 14.9 
dustria 08 3.3 1.7 1.6 2.8 3.0 1.2 9.6 


Bogery 10.) 2.6 1.6 0.8 2.2 1.5 $.2 wl 


Fig. 98. The Relative (or Percentage) Bar-chart. 


values, and the chart is a “‘relative” chart, or chart of relative 
values. Where both absolute, and relative charts are being 
shown together, it is a good practice to use black for the ab- 
solute data and bars and red for the relative data and bars, a 


& 


Tranepor- 


S 
s Manufacturing tation ~ Trade 


29.9 $2.9 


. . . . 3 
57.8 : a9 
i Ze ; ee 
was 2 Fees etn eeee 


Agriculture Monufecturing , Trade 
% 


On, 


WOMEN 


Professional Service Clerical 


OCCUPATIONS OF THE EMPLOYED 
Percentages of Population. 10 Years Old and 
Over Engaged 6 
Grttcanieeeee gaged in Cainful Occupetions 
1920 


(Source + United States Census) 


Fig. 99. Any Pair of 100% Bars Really Form a Relative Bar-chart. 


practice to which the ordinary two-color typewriter ribbon and 
carbon papers easily lend themselves. 


COMPOSITE BAR-CHARTS 115 


A very different kind of chart is the one for which we sug- 
gest the name of multiple bar-chart. Here, two or more series 
of values, which are in themselves all totals, have been inter- 
larded or dovetailed and fitted into each other and the bars 


KEY TO SCALES FOR CHARTS 


Rxporte Imports (Sonle of Millionn of Dollara) 
(Dollars) (Dollars) 
1 2 $ 
TOTAL €,080,480,821 5,278,461,490 
Crude materials for use in manufacturing 1,870,767,064 1,751,893,014 


Foodstuffs in crude condition and food animals 917,990,828 577,626,948 


Poodetuffs partly of wholly manufactured 1,116,605,173 1,238,138,941 


Manufacturee for further use in mnufacturing 958,496,878 4 802,503,406 


Manufactures ready for consumption 3,204,857, 759 876,725 ,060 


Miscellaneous 11,765,129 31,694,121 


PORBIGN TRADES OP THE UNITED STATES 
Claseified by nature of artisles 


1920 


Fig. 100. The Multiple Bar-chart. 


should be carefully and clearly distinguished in color and 
shading to show to which series they belong. The bars should 
generally be made narrower, to fit them closely together, into 
little groups, one group for each stub in the data. The data 
on the chart can still be prepared in distinct columns, although 


PUBLICATION OF NEW BOOKS 
The leading nations 
1919 & 1920 
(Souroe:- Le Droit d'Auteur, Paris) P 
(Note:- Japanese "books" not strictly comparable) 


Bumber of books 


1919 1920 0 5000 10,000 20,000 30,000 
[ ren | 

Germany 26,194 

shgland 8,622 


United States 6,422 
France 5,561 
Italy 6,066 
Holland 3,746 
Denmark 4,465 
Luxemburg 65 


Japan oe 


Fig. 101, A Good Comparison of Historical Data, 


hs CHARTS AND GRAPHS 


the bars have been interlarded, for the data is more easily 
consulted when not interlarded but kept in distinct columns. 
It is really a case of bringing two or more sets of bars together 
for cross-comparison. The result is rarely entirely satisfac- 
tory, and at best is confined to the combination of two sets, 
three or four sets being hard to make mutually distinct, and be- 
coming confusing to the reader. 

In fact, it can be laid down as a general rule that both the 
compound and the multiple bar-charts are too elaborate and 
complicated. A chart is always better the simpler it is, and we 
should make strong efforts to simplify these charts, and if 
possible reduce them to simple bar-charts. It usually pays 
well for sacrifices we make in this way, in legibility and interest 
to the reader, and after all, the chart of this type 1s generally 
directed at a reader, rather than at the maker. The only one 


100% R 


PA 
{__RATIO OF GOLD TO NOTES EXCHANGE IN PERCONT OF PAR a 
JAPAN it 
‘ SWITZERLAND 
‘ ——— a : 
: SPAIN 
‘ Re ere DENMARK SET 
H CME SWEDEN } 
‘ U. KINGDOM ee { 
5 EEE NORWAY ES ‘ 
H (EHS FRANCE ‘ 
H ITALY 26) a 
: BELGIUM ED 
i FINLAND 
¢ B PORTUGAL ' HEY 
‘ 1 GERMANY & 


Fig. 102. Correlation is Indicated by Mirroring. 


Ratio of gold reserves of Central Banks to paper currency in circulation compared 
with the relation of exchange rates to par value (March, 1922)—Permission of 
Mr. Carl Snyder. 


of the three which stands out as absolutely simple and clear 
is the relative compound bar-chart, which consists of nothing 
more than a series of 100% bars. 

The reason for the simplicity of the relative compound bar- 
chart is to be found in its uniform length of lines or bars. Only 
the segmentation of the bars changes in the chart, and the 
reader is not called upon to judge at the same time of various 
lengths and parts, but only of the various parts. In order to 
secure something of the same symmetry that marks the rela- 
tive form, many chart-markers prepare the absolute form of 


COMPOSITE BAR-CHARTS Ut 


compound or segmented bar-chart in a pyramidal or bi-lateral 
form, the individual bars being aligned, not with even left 
hand ends, but with even centers, and each bar extending 
equally to right and left of the center line down the middle of 


1860 


{perio 


SIXTY YEARS OF IMMIGRATION 
Total Number of Immigrants Arrived 
United States 
1660-1920, by years 
(Meximum: 1,285,349, in 1907.) 


Fig. 103. Symmetry has Only a Popular Value. 


the chart. Even simple or unsegmented bar-charts are some- 
times arranged in this way. The form is certainly more pic- 
torial and decorative than the forms which have been described. 
But data is, of course, not easily attached to this chart; in 


118 CHARTS AND GRAPHS 


fact it is ordinarily omitted. When plenty of space for the 
chart is available, and data is not going to be shown, but 
extremely pictorial or sensational effects are desired, this 
symmetrical form would appear to be entirely Anis ble 
But it is certainly not to be recommended under any other 
conditions. 


ae 
a ape i 
cy ye se GER 
| AUSTRI OTHER] AME |ASIA 


=e 


AUSTRIA 


A CENTURY OF I¥MIGRATION 
Countries of last permanent residence of Immigrants Arrived 
United States 
1820-1920, by decades 


[russia | 


Fig. 104. Connection Lines or Shadings to Distinguish Segments. 


A common practice in the elaboration or decoration of bar- 
charts both simple and compound, particularly when they 
represent connected data, is to draw connecting lines across 


COMPOSITE BAR-CHARTS 11g 


intervening spaces between corresponding division points on 
adjacent bars. In later chapters this process is dealt with 
fully, and the omission of bars described, but we may here 


1879 favs 


(26 


‘® X24 WN 1970 


® WEEE 


NAG] 


&9 


Y 1900 
DD SWESSN 


¥. 
*68- earner, clase- Parm Farsers 
ified laborers 


CLASS ALIGNMENTS OF POPULATION 
United States 
1870-1910, by décades 
(from data derived from Census by A. A. Hansen) 
(Hote:- Figures show percentages, bars show quantities) 


Fig. 105. Warping the Chart to Show the Trend of Changes, 


120 CHARTS AND GRAPHS 


observe the effects where the bars are kept and the connecting 
lines drawn in. The reader is, of course, enabled to identify 
and compare corresponding segments of the bars more easily. 
In an extreme form of this chart, sometimes called the “stream 


SOURCE OF IMPORTS 
EUROPE NO. AMER. SQAMER. OTHER | 


1913 


nor - = -» | 
a 


DESTINATION OF EXPORTS 
EUROPE NO. AMER. S.A. OTHER 


1913 
1918 
1919 
1920 


Fig. 106. Connection Lines. Note Inserted Data. 


Percentage of total imports received from different continents, and percentage 
of total exports shipped to different continents——Permission of Mr. Carl Snyder. 


chart,” the bars are broken or bent so that corresponding seg- 
ments are kept as close as possible together. The chart is 
adapted only for data in which the changes of segments are 
fairly uniform and the result is entirely popular in its appeal, 
having no research value at all. What will be later described 
as a smoothing process, takes place in the segmenting division 
lines across the bars, and they, together with their connection 
lines, are made as nearly as possible straight rather than 
zigzag or rectilinear lines. There is little to be said for either 
the symmetrical bar-chart, the stream bar-chart, or the con- 
nection lines between the bars, but they are here described as 
examples of the modification and variation to which the bar- 
chart itself is susceptible. 

It would, of course, be possible to go on and still further 
combine compound and multiple bars until we have acompound 


COMPOSITE BAR-CHARTS Lor 


BOOKS PUBLISHEO IM THE UNITED STATES AND ENGLAND 


New books, new editions and pamphlets published in 1920 
QOmpared as to subject, tor the two countries. 
(Sources* fhe Pudlishers’ Heekly, ¥. Ie) 


TOTAL 


PutLOSOPHY 
Revigion 
Bocrovocer 


d 
tev 


4 

Eovoatios” 

a 
Purrorosy 
Sorewce 
TECcHNOLoeY 
MEOIOINE, HEALTH 
AGRICULTURE 
Domestic Econowr 
Business 
Fone Ants 
Music (woRKS asout) 
Gawes ano Sports 
Livemarure 
Poetry ano Drama 
Fiction 
JUVENILG 
Wistory 
Geoanapuy, Traver 
Brcamapny 


MISCELLANEOUS 


New New Pawo ToTar 
Books Eni TIONS PHLETS 
(American in preso type) 
(British in ttalio type) 
Gum CH 
6,101 1,086 2,235 8,422 
7,675 2,266 41,063 11,004 
1,000. 1,200 
209 33 32 274 
233 31 12 276 
467 37 161 665 
§73 7 29 679 
353 43 363 «759 
637 61 222 870 
70 39 57 166 
213 76 74 363 
101 10 123 234 
164 17 72 253 B 
141 54 49 
176 26 3 
182 49 281 
411 94 92 
259 93 183 
437 155 (128 S 
132 75 83 
269 123 54 
49 18 223 
146 33 39 
22 6 21 
57 13 3 
144 24 78 
103 16 19 
94 6 30 
158 16 10 
44 6 23 
59 6 - 
50 10 62 
120 33 8 
248 53 50 
299 52 15 
409 44 105 
436 80 4? 
778 345 31 
1,088 1,051 15 
410 67 22 
613 148 9 
603 36 172 
438 44 43 
144 22 56 
424 71 109 
271 14 29 
340 33 1 
21 3 ll 
181 - ° 181 


| 


Fig. 107. A Compound Multiple (Absolute) Bar-chart. 


multiple bar-chart. 


Rarely you may have use for such an 
animal, but it really lies out in that field of freak charts into 
which the enterprising chartist will inevitably wander by 
himself, and from which he will surely return if he keeps his 
senses. The field is wide open and there is unlimited oppor- 
tunity for originality in the making and dressing up of bar- 
charts, but in the last analysis all that matters 1s to tell a story 
and tell it well. . You will generally find that this object is best 
attained with the simpler, sounder methods which have been 


122 CHARTS AND GRAPHS 


SEX OF EMIGRANTS AND IMMIGRANTS 


Immigrant Aliens 


Admitted and Emigrant Aliens Departing 
United States 
1917-1920 


(Source: - Report of United States Commissioner General. of Immigration) 


Women (Scale of Percentage) 
Percent 19 20 30 40 50 60 70 80 90 100 


Admitted 


cet Departed 


Admitted 


i918 Departed 


Admitted 


330? Departed 


Admitted 


ieee Departed 


Fig. 108. A Compound Multiple (Relative) Bar-chart. 


here discussed. In bar-charts, perhaps more than in any other 
form of chart-work, we must keep the purposes of simplicity 
and clearness always in mind, and avoid the more complex 
details which will suggest themselves, insidiously and attrac- 


ONITED STATES 


New England 

Middle Atlantic 
East North Central 
West North Central 
South Atlantic 
East South Central 
‘West South Central 
Mountain 


Pacific 


(Seale of ercent) 
perot.0 10 20 50 40 50 60 70 80 90 100 
2S ES Ee een ee es 


URBAN POPULATION 
United States 
1920 
Percentage 


Fig. 109. The Compound Relative is the Best of the Composite 


Bar-charts, 


COMPOSITE BAR-CHARTS 123 


FOREIGN TRADB OF THE WORLD 
Combined Exports & Imports of Leading Nations 
at par of Exchange 
(Source:- United States Statistical Abstract) 


Total Trade 
with 


(Millions 
of Lollars) 


World (40 nations) 75,311 14,479 (uiliions of Dollars) 


United States (1920) 13,259 13,359 
United Kingdom (1920) =1&,926 3,123 
Canada (120) 2,304 1,256 


Frence (2919) 7,429 1,686 


Italy (1919) 4,189 =~, 516 


Netherlencs (1919) 2,639 316 


Japan (1919) 2,421 1,420 


Cermanty (291s) 4,966 877 


Fig. 110. The Simpler Forms Are More Effective. 


tively to us as makers and designers. A simple chart which is 
read and understood is better than a complicated one which 
no one deciphers. : 


CuapTer XIII 
PICTORIAL BAR-CHARTS 


For purposes of publicity, the circular form of chart has 
decided advantages over the rectilinear chart. This has been 
gone into in the chapter on pie-charts. The circle attracts the 
attention even of casual readers. And circular-shaped charts 
are therefore popular with all those propagandists who seek, 
by sugar-coating their information, to dispense it to an un- 
willing and indifferent public. The charts are useful for, and 
should be only designed for, advertising, and the popular pres- 
entation of educational matter. They are useless for research 
and study. ‘These considerations have been discussed in the 
chapter on pie-charts, but arise again in connection with the. 
possibilities of converting series of bars, that is, bar-charts, 
into series of circles. 

Truly, when a series of circles are to be used in a chart, the 
chart-maker’s road should be marked “‘Warning: Dangerous 
Curves Ahead.” For the path he must pursue around the 
unshakable fact that a circular area has two dimensions, is at 
times devious and hard, in view of the rule against showing 
one-dimension data by two-dimension charts. The data 
which is shown upon bar-charts has but one dimension in the 
sense of the rule and the bar-chart itself has but one dimension. 
But when the bar-chart is converted into a series of circles, 
the result is extremely likely to have two varying dimensions 
and be as disastrous as the use of squares discussed in the 
chapter on dimensions. . 

We have seen that the substitution of a single circle for a 
single bar (or 100% bar) is harmless, for the reason that the 
area of the segments of the circle vary directly with the arcs 
and subtending angles. In short, in the pie-chart the areas 
of slices of the pie vary directly with the linear or one-dimension 
variations, and there is no conflict of measurements. The 
chart is like the 100% bar, for that too has an area which varies 


124 


PICTORIAL BAR-CHARTS 126 


directly with one of its linear measurements. In the bar the 
width is constant; in the circle the radius is constant. 

But a series of bars of different lengths can not rightly be 
turned into a series of circles of different circumferences. In 
a series of bars the widths of the bars can be kept constant 
and hence their areas can be made to vary directly with their 
lengths. But in a series of circles of various circumferences 
the radi cannot be kept constant, but must vary with the 
circumferences, so that the areas of the circles will vary by 
the squares of the variations of the circumferences. In short, 
the moment you use circles of different sizes, the old conflict 
between area measurements and linear measurements creeps 
in, the most fundamental principle of charting is violated, 
and the chart becomes fallacious and deceptive. 

There is but one type of bar-chart in which the bars can 
be turned into as many circles, and that is the relative com- 
pound bar-chart of the last chapter—a chart which is nothing 
more than a series of 100% bars. As neither the total length 
nor area of these bars varies, they can be safely turned into 
as many pie-charts or 100% circles of uniform circumference 
and area. The chart is one in which the segments alone are 
significant. It is true, as was said in the chapter on pie- 
charts, that the segments cannot be so well compared as in 
the relative compound bar-chart, and for this reason the chart 
is of less value for careful study, but the circular shapes have 
been secured and the chart has been perhaps made more at- 
tractive and popular. : 

For the simple and multiple bar-charts there is a dodge by 
which circles can be used, if you are intent on circles at any 
cost. The result is not appreciably less interesting and it has 
the advantage of being accurate. It consists in using whole 
circles and fractions or fragments of circles, all of uniform 
radii. Adopting one value in the series—perhaps the average 
—as 100%, you must turn all your data into percentages 
before preparing this chart and then plot as many circles and 
fractions of circles as the data calls for. This method of 
charting is sound because throughout the circles and fragments 
of circles, a uniformity of radii has been maintained, and the 
areas vary directly with the circumferences and arcs. 

The drawing of these charts is comparatively easy, as all 
circles and parts of circles can be put in with a bow pen or 
compass without changing its setting. The work involved is 


126 HIGHEST PRICES OF FOOD 
Index Numbers of Retail Prices 
United States 
Average 1913 * 100 ‘ 
(Source:- Bureau of Labor Statistics) 


#11 articles (Jun, 1920) -@e@+ 


Plate-beef (Apr, 1919) 
Chuck roast (May, 1919) 
Bacon (Jul, 1919) 
Lard (Jul-Aug, 1919) 
Cheese “(Aug, 1919) 
Butter (Deo, 1919) 
Coffee (Jan-Jul, 1920) 
Hens (Apr, 1920) 
Rice (May-Jun, 1920) 
Plour (Jun, 1920) 
Potatoes (Jun, 1920) 
Sugar (Jun, 1920) 


Sirloin steak (Jul, 1920) 


Round steak (Jul, 1920) 
Rib roast (Jul, 1920) 
Corn meal (Jul, 1920) 
Bread (Jul-8ep, 1920) 
Toa (Jul-Sep, 1920) 137 
Fan (Aug-Bep, 1920) 224 


Pork chops (Sep-Oct, 1020) 238 


“Milk (Oct-Nov, 1920) 194 


Bege (Dec, 1920) ~ 268 BOo® 


Fig. 111, The Circles Must Have Uniform Radii, 


aa 


PICTORIAL BAR-CHARTS Oe 


far less than it would be if circles of different radii had 
been used, requiring fresh setting of the pen for each circle, 


FOREIGN TRADE OF THE WORLD 
Combined Exports and Importe of Leading Nations 
at Par of Exchange 
(Source: United States Statistical Abstract) 


© (Millions & 


of Vollars) 


World (40 nations) 78,312 14,479 


(Millions of Dollars) 


nites ster 800) 33899? OOSSOSCSOSSOSO” 

United Kingdom (1920) 15,925 3,123 G@QOOOCOOCOCOCOOD) 
Canada (1920) 2,304 1,256 ee’ 

France (1919) 7,429 1,686 @SRQ000CKCL° 

Italy (1919) 4,189 1,516 @acn™ 

Netherlands (1919) 2,639 316. SOs) 

Japan (1919) 2,421 1,420 eer 

Germany (1913) 4,966 577 — 0083) 


Fig. 112. Segmented, Like the Compound Bar-chart. 


from square root calculations of the variations. ‘The seg- 
ments or fragments of circles can ordinarily be’ drawn in at 


ale) s 


= 
Misoellanecags Foreign Orgenieed Individual Religious Health Education Fine Arts Roe and 
Reforn Relief Charities Personal Purposes Recreation 
Organizations Gifts 
0.57% 9.6% 11.36% 11.42% 47.75% 9.04% 8.0% $.14% 0.86% 


HE IDEAL PHILANTHROPIC BUDGET 
a Proposed National Budget of Philanthropic Donations 
The United States 
1921 
Total $1,749 1009, 000 * 
(Source:- Peul and Dorothy Douglas, What Cen a Man afford?") 


Fig. 113. Suggesting Metal Coins. 


CHARTS AND GRAPHS 


128 


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PICTORIAL BAR-CHARTS 129 


sight, very accurate work for which protractors would be nec- 
essary not being of any value in this chart. 


This form of chart is largely an attempt to present bar- 
chart information popularly. It is for that purpose particu- 
larly adapted to financial data, in which the circles can be 
taken to represent dollars and the fractions of circles parts of 
dollars. While in strict theory the circles should be at even dis- 
tances from each other, yet where there are several for a single 
bar or figure, the conception of metal money is so vividly pre- 
sented that the circles can be overlapped. The overlapping 
or circles saves much space without lessening greatly the im- 
pression onthe reader’s mind. It is as if, in the West where 
silver dollars are still used, you should lay out a row of these 
coins, each, except the first, tilted up and resting partly on 
the next one. But where space does not require this crowding 
up of circles, it is better as a general rule to place them at 


462,000 544,000 


AUTOMOSILE PRODUCTION 
Number of Passenger Cars Produced 
The United States 
1913-121 
(Source:- National Automobile Chamber of Commerce) 


Fig. 115. The Third Dimension Is Ornamental. 


CHARTS AND GRAPHS 


130 


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PICTORIAL BAR-CHARTS L1 


even distances. They will then roughly form bars of circles 
or coins. 


Nor do you need to present a row of circles in the place of 
your bars. Rows of human figures, all drawn to the same 
scale, can be used in the place of bars. The length of the rows 
and the number of figures depicted in each, will show quite 
as well as plain bars would have shown, the various amounts 
represented, This is a method which those who wish to com- 


PROLUCTION CF BASIC COMMCLITIES 
Index Figures cf Monthly Preduction in Specified Industries 
United States 
Dec. 1921-Jan. 1922 
(Source:- Federal Reserve Bulletin) > 
(Normal « Trend after Allowing for Seasonal Variations and yeareto-year growth = 100f 
(Black Arrow = Jan. 1922) 
(White Arrow = Dec, 1921) 
(Dotted Arrow s Low of 1921) 


Fig. 117. Pointers, Instead of Segments, Suggest Pressure Gauge Dials, 


132 CHARTS AND GRAPHS 


THE HISH COST OF LIVING 
Index Figures of Xetail Prices 
United States 
June, 1920 
(Source:- Monthly Labor Keviewr) 
(1913 Average ® 100) 


Fig. 118. Aeroplanes, Horse-races, Boat-races, and the Like, Have a 


Certain Popular Value, 


FURNITURE AND 
293 FURNISHINGS 


—|288 CLOTHING 


214 FOOL 


1202 MISCELLANEOUS 


172 FUEL AND 


LIGHT 


135 HOUSING 


100 ® 1913 PRICES 


{ 
° 


PICTORIAL BAR-CHARTS 133 


pare pictorially two or more populations, can safely employ. 
Instead of showing the Japanese army with a single small 
soldier and the American army with a single large one, in 
which case you confront the reader with three conflicting 
measurements—height, surface area, and cubic volume or 
weight—you need merely show one Japanese soldier and sev- 
eral American ones, all of the same size, and their number will 
give an accurate conception of the relative sizes of the two 
armies. 

Here, then, is the answer to the problems raised in the 
chapter on charting principles. Here is the proper way to 
show pictorially the comparison between two or more items. 
Do not draw one loaf of bread and an enlarged replica of it 
beside it, to show how much the food-bill of the nation has 
changed, but draw one loaf of bread and label it with the 
earlier year and draw several loaves of the same size and label 
them for today. Do not bring together a large and a small 
house nor a large and a small nugget of gold, nor a large and a 
small railroad car, but place together a single one of each and 
a group. The number of times this simple rule is violated, 
with results which vary between gross understatement of a per- 
fectly good case and gross deception about a poor one, will 
amaze you when you begin to watch for it. And the amount 
of money spent sometimes in publishing them, futile or false 
as they are, will also amaze you. It is one of the most frequent 
of all errors in charting. ’ 

In addition to the geometric pattern of the rectangle and 
circle, there are countless pictorial devices, the simplest of 
which is to indicate a third dimension to the bars, setting 
them up on end for this purpose. The ingenuity of adver- 
tising artists has hardly been tapped as yet, and thermometers, 
barometers, or pressure-gauge dials are but the beginning of 
the avalanche. The pictures of motor, horse, or boat-races, or 
altitude flights of aeroplanes have already been found useful, 
and it is probable that all popular contests can be made to 
yield attractive pictorial substitutes for the prosaic bar-chart. 


CHAPTER XIV 
VERTICAL-BAR CHARTS 


Between the sensational picture-bar which we have just 
considered, and the plain bar itself, there is a type of bar 
which is both accurate and popular. Of less value in the stat- 
istical laboratory, it nevertheless deserves a passing glance 
even from the most academic investigator, for it forms an 
interesting link between bar-charts and higher things. 

To make a bar-chart popular, knock it over flat on its side, 
so that the bars stand up on end. Simple, isn’t it? But that’s 


foe Yorn |: 


Peoasylvasio | 


Mitaots | 


Obie |: 
Gasoreraserte |: 


Bow Jersey | 


Mienigia |) 
Celiferate 


PRODUCTION IB THE UNITED sTATEO 


Distrivation of Gress Value of 
Meurtaotared Products, shoving 
Swolve leoding othtes, 1019, « 


Fig. 119. 


’ 


the rule. There being nothing more to discuss in the matter 
of making popular bar-charts, we are tempted to close the dis- 
cussion at this point and produce a pleasant surprise to all. 


134 


VERTICAL-BAR CHARTS 135 
But the vertical bar-chart is rich in suggestions for the higher 
forms of charts which we are approaching, and it deserves 
a close study. | 


Fy 1918 1981. 
$3, 181, 406,000 $5, 949, 674,000 $6, 035, 580,000 


GOLD RESERVES OF TBE BORLD 
Bonk Holdings, 1018-1981 


Fig. 120. 


The chief value of the “‘pipe-organ chart” as it is sometimes 
called, lies in the realistic picture it gives of quantities. From 
a base line these quantities are seen to rise the full length of 
the bars, as so much substantial material stacked neatly in 
piles where we can compare them. We view them from the 
‘level or floor on which they are piled. We do not have to 
climb up and get a bird’s-eye view of them as in the ordinary 
bar-chart, where we seem to be looking down upon rows and 
rows of goods, but we see them from a natural view-point. 
Nor do we rely upon an arbitrary arrangement by which their 
left ends have been brought together as in the bar-chart, but 
we know instantly that if they are piled up, it is their tops 
which we must watch. The pipe-organ chart finds instant 
response in our minds, and appeals to us as both logical and 
natural. A child can comprehend it. 

If you call this base-line the x-axis of your paper and give 
the upright bars values in the y-axis, you will be reminded of 
co-ordinates and maps. But it is not necessary to go so far. 
Merely think of your own back yard, and the nice high fence 
about it which you have just white-washed. Assume that 
through some weird freak of carpentry you built it with 


136 CHARTS AND GRAPHS 


WHEAT TRON OATS COTTON OIL CORN AUTOMOBILES 


ONITED STATES PERCENTAGE OF THE WORLD'S PRODUCTION 


of specified commodities 


Fig. 121. 


boards which run horizontally. Or turn and look at the wall 
of your house, with the weather-boarding running horizontally 
about it. Against such a wall let us pile your quantities in 
neat columns or let us stand up some dark boards of the right 
height against it. You are then ready to take a photograph 
which will be a good pipe-organ chart. The lines of the 
weather-boarding on the house will make the field of the chart, 
and the upright dark boards will be the bars. 

Note also, and this is important, that if through standing 
too close you should take a picture showing only the upper 
ends of the upright boards, but not their full lengths, you 
would consider the resulting picture not only a failure but 
actually deceptive. In other words, you must not omit the 
zero-line or base-line. While you would succeed in showing 
the variation of the top ends more clearly you would no longer 
have comparable lengths. One board might be but a tenth 
longer than the other, but by cutting the lower eight-tenths 


VERTICAL-BAR CHARTS 137 


out of your picture, it would appear to be twice as long. The 
thing simply could not be done, unless you wilfully undertook 
to deceive yourself or someone else. The conception of the 
pipe-organ chart is sound and fundamental. It is perhaps the 
most direct charting method we have. It is almost fool-proof, 
which is more than can be said of most charts. And it estab- 
lishes clearly the vital principle not to omit a zero-line. 

Moreover in the pipe-organ cr vertical-bar chart, we first 
encounter labelling or data difficulties. And if there is one 
motto which we should like to print at the bottom of every 
page in bold-face type, as do the publishers of other valuable 
reference-books, it is this: ““Never separate your chart from 
its data.”’ On the contrary, incorporate the data in the chart. 
For a chart without its data is a poor lost thing indeed. And 
the unhappy reader wishing to know what it means must hunt 


-PATAL INDUSTRIAL ACCIDENT RATES 
for specified industries 
United States 
1913 , ; 
(Source:-0. S$. Bureau of Labor Statistics, Bulletin 157) 
Bates per 1000 workera 


75 aa 


| 

25 EB 

75 
<0? (| 


1.40 ESSAI 
| 


23 
2.00 
3.00 
3.00 
2.40 Fam 
2.25 
1.85 
1,60 


om Q 2 
3 ie e vg 
5 : tan a» 
Oo S -« 8 ° a 
= G Oo oa & ° Si er 
ey o-w © Osta £8 0°55 68 © 
4 é Re ter eh et Cae mee pS) 
Cae as 3 Sean he or ie es 
Z 2 8 Can Ae Way Ynaeee tee 
os & 8 Phe ee et cg erie es 
=> o -« So ON Omg ey OU Orme et A ae 
J > A © Se Ole Pe)” iy tena 
© en es 85 0, Se eae tie yest 2 3 
a w far viet Ae greg Init esie: tt en tee urs SO 
Ss iors (a Bo TS. Ol et. eC wi (Ory Ot i> 
Bir Some ee SO Teo iB <3 ¢ 2? + ar = 
Ou unretn ©." 0 ~- eo Sw Bo Sees St iis ke 
CTS a Ona ee vie Sine a ett ee ee ey Sumlor cms roam c oy 
= ad AL wae OP calm disuse wee) Mria patare Wide bar e eee (eek 
ol eo Ww bk & & Oo ot © 10 er Kio 
a ei Le Ce Syn de Ol. iy SeMk MOMMIES C07 Oy 19 ts 
Pa eden CMTE a Se rE RRC A Sam it: iS AE Pee i a 
2 76). O.. eC OS pier a Se ee ee Sg ee ee ae 
ee OL te rt eG ae oma oi ca ee a 


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138 CHARTS AND GRAPHS 


and hunt and hunt till he locates the particular information 
in some distant table. As a matter of fact, he won’t do it, 
for before he has found his data he has lost his interest in 
the matter, and then what good is your chart? 


In the pipe-organ chart, however, it becomes difficult to 
append data directly to the bars. Following your rule of 
tipping the horizontal bar-chart on end, you would naturally 
have the data down below the bars, reading upward laterally. 
This is at once a logical and a sound place, for the bars should 
be in line with their own data. But because the vertical bar- 
chart is for popular consumption, and because the average 
man does not care to crane his neck to one side and read on 
edge, objection is often raised to this method of disposing of 
the data. 


Rattle 
Goathe 


Dinense Battle 

depths debths 
A i A oe Steer 

deaths 


Date 1846-48 1864-66 1861-65 1870 1898 1899- pee 1904-08 1913-18 


tar Wertoan Crimean Wer Civil War = Pranco- —-Spanieh- Russo- World War Bate 
War Prussian Amerioan. Tea Japanese Wer 
Troope ae Britter French Russian Bore Germn French ae British Russian Japanese 0,6.A. seri cten t 
Ak? roope 
v. 
Total Dewth Rete 126 299 41 383 28 80 209 32 “o 49 9 1% 126 Total Death Rate 


Disease Death Rate 110 230 B41 263 65 26 141 2 26 ty 26 “7 


Battle Death Rete 16 oo 72) (2 3) 88 68 . wie % ee 16 Disesse Dooth Rate 


110 Battle Death Rate 


DEATH RATES IN WARFARE 
Bottle and Disease Death Rates per 1000 Soldiers per Year 
In Specified Ware 


818 
(Source;- The Cratiens ee States Bulletin) 


Fig. 123. 


VERTICAL-BAR CHARTS 139 


Nevertheless the method remains proper, and one is almost 

, tempted to say, let the average man learn to crane his neck 
if he wants to check up on our plotting. As a matter of fact, 
the average reader is generally satisfied to know that the data 
is there where he can get it if he wants it, and so does not 
bother to look at it anyway, An occasional figure in which he 


ACCIDENT MORTALITY 
Death-Rates per 1000 Population of Each Age Croup, and Sex 
United States 
1910-1912 
(Source: - Mortality Statistics, United States Ceneus) 


fi; 


Under 15 15 to 44 45 to 64 65 and over 
Men Women Men Women Men Women Men Women 
663 «395 =1.308 2170) =.11.805 2593 2.985 2,800 

Fig. 124. 


is really interested will be read carefully by him in spite of 
its reading upward, never fear. And throughout the whole 
field of charts it is of such great value to be able to place one’s 
data or figures out along projected lines from plotted bars, or 
points, that we must adopt the upward reading data in spite 
of its temporary strangeness. It is to be accepted and adopted 
as a proper feature of charting. 

Where the bars are very wide, or the spaces between them 
wide, there may, it is true, be room in which to write the data 
horizontally, in little boxes below the charts. This method is 
wasteful of space and compresses words and figures confusingly, 


CHARTS AND GRAPHS 


140 


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SULVY ALITVINOA INGIIOOV FIVN 


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VERTICAL-BAR CHARTS 141 
generally used as it is not in the long run satisfactory even 
for the average popular chart. 


Two principles can therefore be garnered from the pipe- 
organ chart, first that the base or zero-line should never be 
omitted, second that data should be kept with the chart Ee 


24 


UNITED STATES (im 
UNITED KINGDOM CJ 


510 
239 
201 
145 
i j 


‘JAN.r1 MAY 1 SEPT.1 JAN. 


APR. AUG. DEC. APR. 
I9Zi=— = 1922 


Fig. 126. An Absolute Multiple Bar-chart. 
Volume of Foreign Financing in the United States and in the United Kingdom, 
in millions of dollars (pounds converted at current rates of exchange)——Per- 
mission of Mr. Carl Snyder. 


gardless of the direction in which that data must be written. 
And as you progress further into charts, not only will it help 
you to retain these principles. but it will also help you im- 
mensely to visualize to yourself again and again this chart 
composed of vertical bars, a chart from which most of the 
higher forms have been evolved. 


142 


Por cent of 
Waste 
through 
Outside 
Contacts 


Per cent of 
Waste 
through 
Management 


Per cent of 
Waste 
through 
Labor 


CHARTS AND GRAPHS 


Men's Building Printing Boot and Metal Textile 
clothing shoe trades manufac- 
manufac- manufac- turing 
turing turing 


THE BLAME FOR INDUSTRIAL WASTE 
in specified industries 
(Source:- The Elimination of Faste.) 


Fig. 127. Wide Bars with Data Inserted. 


Percent of 
Waste 
through 
Outside 

Contacts 


Percent of 
Waste 
due to 
Management 


Percent of 
Waste 
due to 
Laboe 


VERTICAL-BAR CHARTS 


aaa a 


es 


mal SI SSSI 


Fig. 128. Connecting Lines Are Often Useful. 


Fig. 129. 
Gross tonnage of world seagoing iron and steel ships in 1914 and 1921 (in millions 
of tons)—Permission of Mr. Carl Snyder. 


CHAPTER XV 
CURVES 


It may not have been a very clever fellow who invented 
curves, but he was assuredly lazy. For he balked at the task 
of drawing vertical bars in the pipe-organ style and he said, 
“Since I am only interested in the ends of the bars, I will 
place a dot where each bar ends, and let it go at that.’’ And 
later when he wished to find the dots quickly, he drew con- 
necting lines between them and, behold, he had a “curve.”” A 
curve can, therefore, be defined as a line passing through the 
upper ends of the bars in a vertical-bar chart. 


1790 3,930,000, 
1800 5,310,000 
1810 7,240,000' 
1820 9,640,000! 
1830 12,870,000 
1640 17,070,000 ° 
18650 23,200,000 
4860 31,400,000 
1870 $8,600,000 
680 60,160,000 
890 62, 900,000 
WOO = _: 76,000,000 
310 ~—- 92,000,000 
2920 105,700,000. 


THB POPULATION OF’ 
{HE UNITED STATES 
1790-1920 


Fig. 130. Here is the Data—Historical. 


Let us step out into your back yard again and take another 
look at the upright boards which in the last chapter were left 


145 


146 CHARTS AND GRAPHS 


standing against the wall of the house. Will it not be an ex- 
cellent plan—if the house is not yours—to drive a nail into the 
wall above each board to mark its height? Then we can throw 
the boards away or let the children play with them. And if 


THE POPULATION OF THE UNITED BTATES 
1790-1920 


© 10 20 30 40 80 860 60 100 


° 
1790 $,929,000 


1800 6,508,000 


1900 76,996,000 


1910 91,972,000 


1920 106,711,000 


Fig. 131. The Ordinary Bar-chart. 


we run a piece of dark string along from nail to nail, we will 
not have any difficulty in following the changes in their posi- 
tions. Here we have a home-made curve. A photograph of 
this piece of string (as long as we also show the ground in the 


70,000 ,000 


60,000,000 
60,000,000 
40,000,000 


30,000,000 
20,009,000 


440 ,000 .000 


i) 


9,640,000 fea 

25,200,000 

1,400,000 

8,600,000 feet 
50,890,000 

2,900,000 

76,000,000 /eaMRMemeer teresa! 


ee oe g 8 
ee ee oe er g 8 
ee ee g 05 ¢ 3 g aioe 
&790 1200 


1810 1820 1830 1640 1950 1860° 1870 1680 1890 1900 1910 1920 


HS POPULATION OF THE UNITED STATES 
1790-1920 


Fig. 132, Vertical Bars for Popularity. 


CURVES 147 


picture) will do quite as well as a photograph of the original 
boards, for we can always imagine the boards running from 
ground to string. The picture will be complete if we run 
laths of uniform length up where the boards have been, that 
the exact position of the nails along the wall may be clear 
when we come to make the next curve on the same wall. In 
a chart these up-and-down laths which serve merely to mark 
the horizontal position of the now invisible bars, are called 
“ordinates” and their distances along the ground from the 
first lath are called ‘“‘abscissae’””—words never to be forgotten. 

You will already have observed the wonderful thing about 


12,870,000 Saas 


8 8 
eg 
= ‘. : 
°o a © 

3° 
= 
2790 1800 1810 1620 1830 1840 1650 1860 1870 1680 1890 1900 1910 1920: 


THE POPULATION OF THE ONITED STATES 
1790-1920 


Fig. 133. A Curve Through the Bars. 


THE POPULATION OF THE UWITSD STATES 
1790-1920 


CSRs eater ied 4588 218 oe cee 8 
Seed, Seat. Gg. 9-68 oe gee 
po rs Ose tan tic tice hie a er act Maite eter. 


0 
1790 1800 1810 1820 1630 1640 1860 1660 1870 1860 1890 1900 1910 1920 


Fig. 134. The Bars Disappearing; the ‘‘Field’’ Appearing. 


148 CHARTS AND GRAPHS 


THE POPULATION OF THE UNITED STATES 
1 19 


3,930,000 
5,310,000 


17,070,000 
23,200,000 
31,400,000 
38,600,000 
50,200,000 


Pl 

is 
awe 

a 

iN 

if 

i 

a 


ee 
\\ 


es 

fee 
RE 
Fl 
Ei 
a 


Poy ec wa = 


So oO o 
Boe Whe uhl wees en Se Sea eee 


Fig. 135. The Evolution of the Curve is Complete. 


a curve, namely that it is easily combined with several of its 
kind upon a single chart. Multiple curves are far better than 
multiple bar charts. A number of curves wiggling across the 
page at the tops of invisible bars are eminently more satis- 
factory than actual bars interlarded. In the first place, com- 
parison of several series of data is greatly facilitated in curves ~ 
because each set has been condensed and simplified into a single 
line. There is no difficulty in comparing values of each series 
with each other. In the second place, such a comparison is 
more accurate in curves because all similar points on various 
sets or series have been brought together upon a single vertical 
line. Had we placed the bars in this way on top of each other, 
the longest one would have wiped out or hidden all the shorter 
ones and in the multiple bar-chart, therefore, each set of bars 


CURVES 


has to be shifted slightly out of position to avoid the next set. 
But when we plot only the end points of these bars, the short 
and the long ones show up equally clearly and can be brought 
together upon their true ordinate lines. In the third place a 
curve is much more easily drawn than a bar-chart. This is a 
most important reason, between ourselves. And, fourthly, to 


~—the reader who understands it (and it is a fact that schoolchil- 


dren understand it, however little the present generation of 
adults may) the curve is less confusing and more easily read 
for its salient points. You will find still other reasons why 
curves are advantageous as you go on. 

A curve cannot, however, always be used in the place of a 
bar-chart, for the line which connects the various points im- 
plies that the data itself can be considered connected. Much 
data can not be so considered. A careful inspection of the 
data will soon show whether it is connected or not, for the 
stubs of connected data always form a variable. In the chap- 
ter on dimensions and variables, the test for variable nature in 
stubs was given somewhat as follows: “can the stubs or items 
be shifted up or down in their arrangement freely or is their 
order naturally fixed by their nature?’ Variability is shown 
by the rigidity of order. 

This limitation of the curve method can be made clear by 
two or three illustrations. We have before us the statistics of 
the population of the United States for each ten-year period 
during the last century. The stubs for each figure of popula- 
tion in this case are: 1790, 1800, 1810, 1820, 1830, 1840, and so 
on down to 1920. Now no sane person would think of ar- 
ranging these normally in any order except from the earliest 
to the latest, or from the latest to the earliest—it would be 
ridiculous to adopt an arrangement such as the following: 
1910, 1810, 1800, 1860, 1920 and so on. Clearly, this is a case 
in which the order of the items or stubs is naturally determined 
by the data itself and these various years can be considered 
as the various values of one variable, namely “time.” The 
data can be charted on a curve. Consider another example. 
In taking a census of the buildings in a certain well-known 
village, the investigators returned reports of the number of 
one-story houses, the number of two-story houses, the number 
of three-story houses, and so on up to 55-story buildings. 
Here again the order of the items or stubs, that is the number 
of stories, is definitely fixed by the nature of these items and 


150 CHARTS AND GRAPHS 


the number of stories can be considered a variable in the same 
way as before and the data can be shown by a curve. Take 


IMPORTS INTO RUSSIA 
1921 
(Source:- Russian Information and Review, London) 


($) 
Total 124,281,000 
Foodstuffs 16,061,000 
Animal products 39,605 ,000 
Timber and seed 504,000 
Earthenware 227,000 
Fuel, pitch, etc. 2,786,000 
Chemicals : 2,032,000 
Metals, ores, machinery, tools 29,184 ,000 
Paper and paper gcoods 3,977,000 
Textiles : 15,206,000 
Wearing apparel, stationery, etc. 13,132,000 
Miscellaneous (including 48,000 1,567,000 


tons of "famine aid") 


Fig. 136. Here is Data not in Series. 


another example. The United States exported to England in 
the year 1920 a large amount of copper, wheat, rubber, auto- 
mobile supplies, machinery and paper. If we were charting 
these exports, it makes no difference whether we show the 
cotton exports before the paper exports or vice versa. The 
order of these items is not fixed by their nature and can be 
arranged in any way we desire. Here, then, is a case in which 
we cannot use a curve but must fall back upon the bar-chart. 
In short, while the bar-chart can be used for all data, the 
curve-chart can only be used for data of which the stubs form 
values or readings of a mathematician’s variable.} 


1A curve or connected line in the place of vertical bars, for abstract or geograph- 
ical data (that is, data of which the stubs are not an ordered numerical series) 1s a 
graphic monstrosity, fortunately not often seen. 


CURVES 151 


IMPORTS INTO RUSSIA 
1921 


(Source:+ Russian Information and Review, London) 


($) 


Total 124,281,000 (Million of Dollars) 


20 


Foodstuffs 16,061,000 

Animal products 39,605 ,000 

Timber and seed $04,000 

Earthenware 227,000 

Fuel, pitch, ete. 2,786,000 

Chenicals 2,032,000 ae 

Metal, ores, machinery, toois 29,184,000 kay Bay ; Aycan 
Paper and paper goods 3,977,000 , HTT] 
Textiles 16,206,000 i ¥ Sie 
Woering apparel, stationery, etc. 13,132,000 SRRERSRRRReRE 
Miscellaneous (including famine 1,667,000 bs 


aid 48,000 tons) 


Fig. 137. No Curve Should be Made with This. 


If we examine the field or background of a curve, we will 
find that it is drawn up according to the principles of Cartesian 
co-ordinates which we have already observed. The reader 
who has forgotten or omitted that weird chapter had best 
turn back to it and read it carefully. In order to understand 
co-ordinates for curve-chart work, you must know that the 
x-axis of your chart is that straight horizontal line along the 
bottom of the chart which we sometimes call the base line of 
the chart, or zero line. The y-axis is the vertical line whose 
value at all points is zero on the x-axis. In the ordinary chart, 
x and y axes are along the edges of the chart, the x-axis at the 
bottom, and the y-axis at the left hand side. In this case 
there is no room on the chart for negative values. It is not at 
all uncommon, however, to have charts which reach over and 
beyond the two axes for the plotting of negative values. In 
still other charts, the true y-axis does not appear at all, the 
chart not showing no zero x-value whatsoever. This is gener- 
ally a case of data in which zero itself is meaningless or arbit- 
rary. A common example of it is historical data, such as the 
first illustration in the last paragraph, where the time values 
commenced with the year 1790—not with the year zero. The 
horizontal lines, and particularly the distances along the 
x-axis, are called abscissae. The vertical lines crossing the 
ends of the abscissae or points on the x-axis, are called ordin- 


oe CHARTS AND GRAPHS 


ates. All points having the same abscissae or values along 
the x-axis lie in the same ordinate or vertical line, and vice 
versa, all points having the same ordinates or values along 
the y-axis lie in the same horizontal line. | 

As we have seen, the curve chart requires data with two 
dimensions, the curve being plotted upon a field which has 
two dimensions. Along the horizontal dimension or x-axis you 
will find the values of the independent or x-variable, generally 
the stubs in your table of data. For each value of this variable, 
that is for each stub in your table, there is a corresponding 
value of the dependent or y-variable, namely the figure in the 
column beside the stub in your table of data. This y-value is 
plotted along the ordinate or vertical line from the given point 
on the x-axis (or abscissae, indicted by the stub) to the height 
upon the y-axis indicated by its value. Another way of express- 
ing this is as follows: In the data for a curve-chart each 
figure to be plotted has two values, one being the value of the 
figure itself and the other being the value of its stub in the 
table. These two values of a figure describe the co-ordinates 
of the point by which the figure is plotted on the chart. 

Not only can the point be plotted from the data showing 
its co-ordinates, but the process can be reversed and the co- 
ordinates of a point can be read from the plot or chart, merely 
by following the intersecting lines through the point to their 
respective axes. For it will be seen that every point on the 
paper has two co-ordinates, one of which is the abscissa or 
horizontal line passing through it, and the other of which is 
the ordinate or vertical line passing through it. The point 
itself is sometimes called the “‘intersect” of these two lines. 
As two perpendicular lines can intersect at one point and one 
point only, there can be only one point described by any two 
co-ordinates. We can therefore locate or identify any point 
by its co-ordinates and the co-ordinates of a point may be 
said to fix rigidly its position. 

If our data tell us that a certain town has a population of 
3,000 persons in the year 1910, we should plot this population 
by moving along the horizontal or x-axis, to the distance or 
abscissa of the year 1910, and then moving upward along the 
ordinate or vertical line through that point, to the height of 
the horizontal line (or abscissa) passing through the point of 
population, 3,000, in the y-axis. The dot, or point on the paper 
which would indicate this town, would be placed at the inter- 


a 


CURVES 153 


section of the ordinate for 1910 and the abscissa for 3,000, 
and the co-ordinates of that point would be “year 1910, popu- 
lation 3,000,” that is “x, 1910, y, 3,000.’ And if we see the 
point of this town plotted upon a chart, we can read from its 
co-ordinates the information that in the year 1910 its popula- 
tion was 3,000, simply by following the two co-ordinates of 
the point out to the axes of the chart. 

The distinction between the dependent and the independent 
variables is important. The independent variable is normally 
formed by the stubs in the tabulation and the dependent 
variable by the corresponding figures, that is, the figures in 
adjoining columns. ‘The tabulation is more or less optional 
however, and for certain purposes stubs and data may be 
interchanged. The distinction between dependent and inde- 
pendent variables goes deeper, and finds its origin in the 
peculiar nature of the data. When the readings along any 
variable are made a basis of classification of data, then that 
variable is the independent one. In general, the dependent 
variable is that one whose values may be said to depend upon 
the values of the other variable. Such dependence need not 
take the form of a mathematical equation or explicit function, 
but is merely a matter of convenience in such matters as the 
classification and arrangement in the statistical table. Our 
chief concern is with the plotting of the data upon curves, and 
the important rule to be remembered is that the independent 
variable should be laid off on thé x-axis and the dependent 
one on the y-axis. The rule is not without its exceptions, but 
these should always be founded upon special considerations 
and in the absence of such special reasons, the rule should be 


invariably followed. 


CHAPTER XVI 
FIELDS 
Most of the good things in this world involve some sacrifice. | 


Curves are no exception. In a curve the direct visible connec- : 
tion between the curve itself and the zero line, or x-axis, is 


; : 7 
sacrificed. As time goes on and you become more and more | 
used to the curve chart, you will begin to think of its values 

| 
f 
Uillions 
Willions 100 | 
190 
90 
90 
80 
80 
At] 
70 
60 
60 
80 60 
40 Ah 
Py) 30 
20 20 
10 10 
° ° 
1960 «61870 «61880 «61890 §=693900 )=—1910 1860 1670 1880 1890 1900 1910 1860 41870 1880 1890 §=63900) =: 1910 
Millions 
100 
90 
80 
70 
60 
50 
ao 
30 


1860 «1870 «1880 2890 «3900-1910 


1860 ©3870 «61860 = 1890) :1900 1910 


Oks ce wisi e/eeM ane ce hesaen eae 
1860 1870 1880 “3690 1900 1910 


From Mr. John Wenzel's ‘Graphic Charts that Mislead," i ienti i 
June 6, 1917. - ‘ e wslead, in Scientific American Supplement, 


Fig. 138. The Amputated Chart is Deceptive. 
154 


pone aa aaa 


PEELDS 0 165 


as in some mysterious manner floating disembodied along the 

connecting line which forms the curve. You will be tempted 
to forget that the quantities rest very substantially upon the 
floor (base line, zero line, x-axis or whatever you want to call 
it), and that it is only their tops which reach the points plotted 
in the curve. And forgetting this, you will try to save space 
by omitting the zero line and lower part of the chart, and by 
showing only that small portion or band of the chart through 
which the plotted curve travels. 

This practice of omitting the zero line is all too common, 
but it is not for that reason excusable. The amputated chart 
is a deceptive one, tempting the average reader to compare 
the heights of points on the curve from the false bottom of the 
amputated chart-field, rather than from the true zero line, far 


Millions 
240 


Millions 
240 . 


R96 485,640 


“| sl ae! | 

SS ae 1 | #30 

yf gece OME | | 

110 Hil 

oe 210 

rso| ORT 

vol _ ST 200 

oo 

eo MPP 190 

nol fT 

roo A PAUUOURUUTIATE 180 

ool PUPUUU TOUR T EE 

olf PUTT UOT 370 

POTEET TET EETTETT | 

col PPUTEU A UUTE ITE 160 

sol RUT | 

ol MEET ET 160 

sol AAUP OUR A CRU EGET 

col POR UUU PROUT EE 140 

MOUUUUROUUOR TOD 

MME - 330 
BESS8 5539 2238283 HRsSE SRSA ERS E EES 
2g 2 3 2 2 2 2 
«a a - rv # ps wt # 


From Mr. John Wenzel's ‘Graphic Charts that Mislead,” in Scientific American Supplement, 
June 6, 1917, 


Fig. 139. The Case Against Amputation is Clear. 


7: = = 7 
f 
7 


156 CHARTS AND GRAPHS 


below and invisible. A curve-chart without a zero line is in 
general no whit less of a printed lie, than a vertical bar-chart 
in which the lower part of the bars themselves are cut away. 
The representation of comparative sizes has been distorted 
and the fluctuations (changes in value) exaggerated. In a few 
more years, the principle that the zero line, when zero is real, 
must normally be shown in a curve will be universally accept- 
ed. Then the emphasis which now must be laid upon this 
principle, will not be needed. Indeed, the author plans in his 
fortieth edition of this work, to omit almost all reference to the 
rule. But, today, you will repeatedly find violations of the 
rule complacently propagating false impressions. And today 
the principle must be iterated, reiterated, and forever kept in 
mind. 


CURATIVE EFFECT OF DIPHTHERIA ANTITOXIN 


Tempereture record of typical case of Diphtheries with prompt use of Antitoxin 
(Source:- U. 8. Public Health Service) 


Degrees pmb te aS Seay Be Me mpi Bees) bet Aah OMe tece tet ey aks: be 
Mireniehé ¢ 2.818 S° Se cs 6. a eee re eee eresa, ues e 
110 
108 
Pele FE ao cl 
106 t+ ++} +4-+ +4 
ra oT Pee Sea on ed BE 
ie cha: | MESA Lc dhe el 
HiME 
Al L1LLaL Tene 
_ Lena 
: suuUomm 
Normal. BBE cot 
re 
96F 
Be mi 
a | Ef 
i ti 
ER 
os ral 
a a 3 a a ee eo ee i er Oe ee eee ly aE Et 
Ba), 26 Delp Samy Tey Patera “aes. cab Gy Se a te tee POP a Ge wer Ga) a Se 
bays; - 1 2 S) 4 6 6 7 


Fig. 140. When Zero is Arbitrary, it can be Omitted. 


There is but one case when the omission of the zero-line 
on the y-axis or dependent variable, is justified. This is the 
case in which zero itself is an arbitrary value, and does not 
really mean a “nothing.” As we have seen in the case of 


; 


FIELDS 157 


x-values in historical data, the year O does not really signify 
zero years, but merely signifies an arbitrary point of time 
from which counting is begun. Science has many such arbi- 
trary zeros; in the Fahrenheit scale of temperature, for in- 
stance, zero degrees is really an arbitrary point. Common 
sense will tell you when the zero is a starting point of the 
quantities measured by your data. And whenever zero is 
really such a lower limit, the rule that the zero line must be 
shown on a chart applies. 

Sometimes, even with the best of intentions, rules must be 
violated and we must do the unjustifiable. The usual excuse 
for amputating a chart is that to show the zero line would 
require too much space, or would reduce the scale and make 
the fluctuations of the curve less noticeable. Sometimes you 
will feel the force of this argument very strongly. It is par- 
ticularly frequent in charts of dividend, interest, and yield 
rates, where the fluctuations are in percentages and the base 
is understood by everyone to be 100%. ‘The argument has 
greater force when the chart is intended chiefly for circulation 
among those in a profession who are already accustomed to 
think of the minor variations of percentages, and who would 
study the chart with interest only with regard to its time to 
time fluctuation-quantities, but would have no interest in its 
relative total quantities. Here it may be argued that the am- 
putated chart would deceive no one and would be of greater 
service than if, at the cost of detail, it were made complete. 

When this argument arises, it must be scrutinized with 
care and hostile scepticism. Often the argument will be found 
specious, resting more on the familiarity of the chart-maker 
himself with his data than upon the true attitude of those 
who will see the chart. In other words the maker of the chart, 
in the thoroughness of his own understanding of the data, 
forgets that others will be less familiar with it, and attributes to 
them his own skill and comprehension. It is easy, in this 
way, to be modest about one’s own powers of understanding, 
but the modesty is costly when, as so often happens, the reader 
is loath to admit his inferiority, and merely lays the charts 
aside for study at some time “later on” which time, needless 
to say, never comes. . 

Only if it is quite certain that no misunderstanding will 
result, should the chart be amputated for the sake of saving 
space or exaggerating fluctuations. And even in such cases, 


158 CHARTS AND GRAPHS 


great care should be taken to make the amputation self-evident 
to the most casual reader of the chart, for it is precisely the 
man who has little time for study of the chart, who is most 
likely to be deceived by it. The best method of making the 
amputation of the chart obvious is to blot out with Chinese 


WORKERS OUTPUT AND FATIGUE 
Index Numbers of Hourly Output in Dexterous Handwork Operations 
United States 
(Source:+ United States Public Health Service, Bulletin Wo. 106) 


Commutator loom 86.2 97.2 97.2 100 95.5 94.3 93.2 86.6 
Magneto taping 92.4 100 95.7 95.1 96.8 96.6 96.0 91.5 
Rell coil 96.1 98.8 100 «98.4 92.2 98.4 98.8 88.3 
Rivet prees 91.2 95.9 100 «694.8 94.5 92.9 94.2 89.1 
Average 94.8 95.6 95.6 88.9 


- 


aed 


a 
. 


TESST 
a 


2 wo re) rte) 0 2° 
d £ & ee Be & x < 
x S is S a “ © oT 
First Second Third Fourth -Bifth Sixth Seventn Eighth 


Fig. 141. The White Zone Warns the Reader. 


white a small irregular zone across the lower part of the chart- 
field, or to erase the co-ordinates in this zone, and show the 
zero-line below this zone. The chart then has the appearance 
of being broken off between the zero-line and the curve, and 
anyone will see that it does not show full distances to the 
curve-points. An easier but less effective method is to make 


FIELDS 19 
$100 


4 
=, 
x LIBERTY BONDS ~~~ 


Pen be 


“ 


6 BRITISH WAR LOANS 


40 CORPORATION BONDS 


1919 1920 1921 1922 
Fig. 142. The Uneven Base-line Indicates That It Is not the Real 
Base-line. 
Average monthly prices of liberty and corporation bonds at New York and 
British War Loans at London.—Permission of Mr. Carl Snyder. 
the lowest horizontal on the chart-field wavy or ragged, again 
with the purpose of indicating that a part of the chart has 
been broken off.! In either case the final object is to remind 


MOM chi Date ee REDE Cae Ts PERE EL ET 
Se 
F SCALEEEPEEDRE ELLE CLEEEEEEEEEEEECE tT 
LINDT LAA | LEN 1 
oo LLL AH MELCHETT 

ua HEE 


120 


nS 


Harvard oe of Economic Bee Rion 
Fig. 143. A Wavy Base-line is a Shorthand Warning. 
Adjusted Index of the Volume of Manufacture (100 = Normal). 


1 The use of rounded or dotted base-lines to indicate abbreviation, which is some- 
times advocated, seems ill-advised, since the method is not self-explanatory and hence 
defeats its own purpose. The object is to flash to the casual glance the abbreviated 
condition of the chart, and any symbolism which must be technically understood is 
of no more value than the scale-figures themselves for this purpose. 


60. CHARTS AND GRAPHS 


the reader than the true zero-line, from which he should meas- 
ure the quantities shown by the curve, lies far below the visible 
portion of the chart. 

Another principle which will quickly appeal to your common 
sense, is the rule that when zero is real, the zero-line should be 
extra heavy to make it prominent. Remember that it takes 
the place of the floor or lower end of the bars in the bar-chart. 
It should stand out, therefore, in such a way that the reader 
can easily grasp its significance and compare with it the 
heights of the points on the curve. The rule is particularly 
important in cases where the chart extends down below the zero 
line into the negative side in order to show negative and posi- 
tive values. On the same principle the 100% line, when it 
occurs in a chart, should be similarly heavy as it also may be 

INCOME CF RAILROADS 
Net Railway Operating Incomes of Class I Reads 
(.0., these having Annual Operating Revenues ia Excess of $1,000,000) 
Unitea States 
1920-1921 
(Source:» Interstate Commerce Commission) 
(Note:- Net railway operating income is total operating revenue lese 


totel operating expense, railway tax accrusls, uncollectitle railway 
Tevenues, equipment. and joint facility rents.) 


Bi 6." OFC SO Aerts fo] S78 8 S02 1S. 2 € 8 i853 os 48 2 
Go ke or Ee © Fo 
e828, S Soe ee Sloe Ge eis ue ere lene 8 8 8 8 
; Keyes SPS ler SSeS rede Ge ee Aen eee cee owe 
Cllrs Gees segues ests F< eeeesggsegze ss Bs 
Kets @o = ee 2 oe o pid © oe oe De a ” o nu °. o x nx are — mn a 
OO. Ge Nar De Tus egitice nox hr Ot Sy PN (Bs, Oem Stk tN ICN be eee oan aed 
OR Ss wae ot ret Os - Bare ao. i 4 1 N44 6) 26.7 @. (Be ee Fe: ee 
' ' ) 2t5 crt . = 
; 
100,000,000 
60,000,000 
© 
50,000,000 
= 100,000,000 
#150,000,000 


1920 1921 


Fig. 144. The Zero-line Should Always be Heavy. 


\ 
q 
4 


FIELDS 161 


considered a base for zero points, being the point of zero loss 
or gain. In fact, the rule may be extended to all cases of lines 
showing significant constant values, and the zero line should 
not be heavy, unless it has a special significance. In charts 
showing temperature in Fahrenheit degrees, for example, since 
the zero point is merely an arbitrary value like any other 
number of degrees, it would be more sensible to emphasize 
the freezing and boiling point lines. Common sense must be 
relied on to determine the lines which can be usefully em- 
phasized. 

The ordinary curve-chart has three different types of 
figures which must be attached to it. These figures are: first, 
the scale figures for the x-axis, showing the values assigned to 
the vertical lines; second, the scale figures for the y-axis, show- 
ing the values which have been assigned to the horizontal lines; 
and third, the data figures, showing the values represented by 
the various plotted points of the curve. There is considerable 
confusion and difference of practice in the positioning of these 
three sets of figures. By going back to the first principles, 
however, and recognizing that a chart is merely a fragment of 
the co-ordinate system of measurement, we can easily find the 
logical and natural places for these figures, and it so happens 
that the positions which are the most logical have proved in 
practise the soundest and most useful ones. 

The scale-figures along the horizontal or x-axis are really 
the values of the independent variable. In your table of data 
they form the stubs or items. As you read in tlie last chapter, 
this independent variable belongs on the x-axis; it should never 
be placed along the y-axis, as is sometimes erroneously done. 
The proper place for the scale showing these figures or values 
of the independent variable, is at the bottom of the chart, 
each figure or value being immediately beneath the lower end 
of the vertical line to which it has been assigned. Do not, 
merely for the sake of ornamentation or decoration, place 
these values at the top of the chart also. Do not box them in, 
each with a little square or circle. Do not make the printing 
unnecessarily large. Do nothing more than is necessary for 
simple clear results. These precepts will save you a great deal 
of time in the preparation of your charts and will save your 
reader much trouble in its reading. It is enough to place the 
figures once for all at the bottom of the chart, forming a scale 
along its entire base. : 


162 CHARTS AND GRAPHS 

The figures which are assigned to the various horizontal 
lines should be placed in a column immediately beside the 
ends of these lines. They form the scale for the dependent 
variable or y-axis. In the case of isolated charts, that is charts 
which will appear singly and alone, this vertical scale is often 
placed at both sides of the charts so that the reader can read 
the values of the horizontal line at either side. For isolated 
charts, there is no particular objection to this practise, though 
it may be a work of super-erogation. But, in the majority 
of cases your charts appear in groups, and often on separate 
sheets which the reader will wish to place side by side for the 
purpose of comparison. Then certainly the two vertical scales 
would be a nuisance; one is sufficient, and it should be placed 
at the left hand side. Indeed, it would be best to make this 
rule universal, namely, that the vertical scale should appear 


PRODUCTION OF AUTOMOBILES 
Number of Passenger Cars and Trucks Produced 
United States 
1913-1921 
(Source; National Automobile Chamber of Commerce) 


° ° 
3) 3 ns 3 
Truoke 8 8 3 ra) 8 3 8 
(en =n a ow ~ - 
. “” wo + °o o oe wo N ” 
~~ I . a i) N a iJ = 
a i nm -” - 
; ° ° ° 
SMe pee: Se remy ent tore ay ire 
Paskenger Ss = e o > o So. 
are a + “ih a 4 i S & m4 
+ wo on a ts) nm o @o wo 
a a SS ar a 


Ler T eT TTT 


400,000 


300,000 


200,000 


PPT TTT 
Lit T 


LLLP 
GROSBRGERESSehEL 


100,000 


\ 


o 


20,000 


1913 
9 

1915 
1916 
1917 
1918 
1919 
1920 
1921 


Fig. 145. The Sound Position for Two Vertical Scales. 


bi 


once only, and then at the left hand side of each chart. The 
use of two scales, one on each side of the chart, is desirable 
only for more popular results. 

When several curves are shown upon the same chart, it is 
often desirable to use different scales for them. That is, the 
same horizontal lines may be given two or even more different 
values for different curves. But even in these cases, it is better 
to place both scales, once and for all, at the left hand side. 
The practise of placing one of these scales at the right hand 
side, and another at the left hand side, has little to recommend 
it. Theoretically, at least, the left hand end of your chart is 
normally the y-axis itself, and the scale or ‘scales should logic- 
ally be attached immediately thereto. In practice this logical 
position is justified. 


PER Cenr. 


1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1av2 15/3 
1913 1914 1915 1916 191T 1918 1919 1920 {92! 


Fig. 146. An Interesting Comparison of Different Periods. 
Here it is not the y-axis, but the x-axis, which has two scales. 


Monthly price index of 14 basic commodities during two war periods. Pre-war 
year in each case is taken as the base of 100%. Prices of the same commodities 
are included for each period —Permission of Mr. Carl Snyder. 


We come then to the question of the third set of figures, 
namely, to the data itself, which the curve represents. As we 
have said before, this data should always be presented with 


164, CHARTS AND°GRAPHS 


the chart.2 It should not be omitted entirely or separated and 
printed in an appended table. That rule is one of the most 
important in chart work, and he who violates it fails to afford 
his reader with convincing proof of the accuracy of his chart. 
We must find a way to insert the data in the chart, and typo- 
graphical difficulties or inconveniences must not be allowed to 
deter us. 

As a matter of fact, the place for the data is obvious. It 
should be placed at the top of the chart, each figure immedi- 
ately above the point by which it is represented on the curve. 
The reader can then glance down the ordinate or vertical line 
through any particular point and find the stub or value of the 
independent variable, and he can glance up the same ordinate 
or vertical line, and find the value of the curve at that point, 
that is, the exact value of the dependent variable. Needless 
to say, he could have found this latter, the dependent variable 
value, with approximate precision, by careful study of the 
horizontal line through this point and its intersection with 
the y-axis. 

In entering figures on the chart for the scales of the inde- 
pendent variable and for the data itself, we come to the same 
typographical difficulties which we met in the pipe-organ or 
vertical bar-chart. We do not often find sufficient room to 
print or write these figures horizontally on the page. Even if 
the chart is so large that we can, by fine printing, crowd the 
figures together horizontally, we will generally find the results 
unsatisfactory. In the first place the figures tend to run into 
each other, and in the second place they lie across, rather than 
in line with the ordinates or vertical lines to which they are 
attached. If we attempt to box the figures in with squares, 
circles, or diamonds, we merely add to the confusion of the 
chart and detract from its simplicity. 


2 Tt is obviously the chart from which data has been omitted, which has led Professor 
Secrist to say: 
“Tabulation of classification precedes; the use of diagrams follows. The 
former generally serves to clarify the meaning of data; the latter frequently 


to obscure it. . . Diagrams alone are more likely to serve as bases for 
conclusions arrived at without study and to foster a disregard for the details 
from which diagrams are drawn . . . Diagrammatic illustrations can 


never replace data themselves, no matter how accurately they tell the truth 
or how illuminating they are. They are at best statistical aids and should be 
so viewed by those who use and study them. "A well-drawn and cleverly 
executed diagram is never a guarantee of the value of the statistical facts 
which it illustrates.’—Secrist, Horace, 4n Introduction to Statistical 
Methods, The Macmillan Company, New York, 1917, pp. 159, 161. 


FIELDS - 


The sound principle, therefore, and one which will be found, 
after a little practise, eminently satisfactory, is to enter the 
data-figures and all except the simplest x-scale figures, vertic- 
ally, that is, by writing on edge. The reader has little diffi- 
culty in turning the page about to read these figures when he 
wishes. ‘There is, therefore, no great disadvantage to this 
method. The figures lie clearly along the lines of the ordinates 
to which they belong, so that there is no doubt or confusion 
in finding the figure for any particular point on the curve. 
You will notice, moreover, that the figures for the independent 
variable and the figures for the data arrange themselves in 
the familiar form on your original tabulation, the only differ- 
ence being that a chart has been inserted sideways between 
the stubs and data. And this is logically sound, because the 
curve is merely a modified form of the pipe-organ chart, and 
the pipe-organ itself is merely a bar-chart placed on its side. 
Even the column headings are retained in the curve chart in 
their same relative positions to each column of figures. 

You will find it useful to keep this relation between the 
curve-chart, the pipe-organ or vertical bar-chart, and the bar- 
chart proper or horizontal bar-chart, always in mind. Par- 
ticularly so, when you have several columns of data to be shown 
by several curves upon the same chart, for in this case it is 
important to retain the column headings at the top of each 
column of figures in the data. These column headings will 
then be to the left of the chart itself and the only difference 
will be that they can be written on horizontal rather than 
vertical lines, so that they can be read easily while the curve- 
chart is in its normal position, though the data is on edge. 
You will, however, find it useful to re-arrange the order of the 
columns of data so that the position of each corresponds 
roughly to the position of its particular curve, the data for the 
uppermost curve being at the top, and the data for the lowest 
curve being at the bottom of the series of data on the chart. 
In order to distinguish two curves on the same chart, which 
may or not cross each other, you will probably use different 
colors for these curves. In that case, it is useful to observe a 
similar. color distinction in the printing or typing of the data 
columns, each column being printed in the color in which its 
particular curve is plotted. You will also find it useful to place 
a small sample section of this curve immediately beside or 
underneath the column heading for the data to which it is 


r, - Sees. Pee 


166 CHARTS AND GRAPHS 


attached, thus forming a sort of key to the curves used on the 
chart. . 

For those who use typewriters (and in all large offices, it is 
well to use typewriting exclusively for the lettering and figure- 
writing on charts) the arrangement above described for the 
positioning of figures and data, will be found extremely con- 
venient. It gives the typist no more trouble than the prepara- 
tion of an ordinary table or tabulation of figures. Placing the 
chart sideways in the typewriter, she types in at the left hand 
edge of the chart-field the stubs or items of the table, and at 
the right hand end of the chart, the figures of the various 
columns. If the ordinates of the chart have been arranged at 
the precise typewriter distances of §, 3, or 4 inch apart, she 
has no further adjustment of the paper to make in the type- 
writing machine. In line after line down the page, she merely 
reproduces the table of the original data leaving a wide gap 
between stub and columns of data—a gap which is filled up 
by the field of the chart. The chart can then be plotted in 
upon this field after the data has been typed. 

The whole process of making a curve chart takes no more 
time than that of making a bar-chart, and in fact, very little 
more than that of making a plain mathematical table. The 
only instructions the typist must have are the data (which can 
be in the form of the original table) and clear orders as to (1) 
which columns must be copied on the chart, (2) the order in 
which they must be placed, and (3) the color, if two colors 
are used, in which they must be typed. The only instructions 
which the draftsman needs are then contained in the form 
itself on which he is to draw, his instructions being the data 
as already typed on the chart. If the vertical-scale figures for 
the y-axis of the chart have also been typed in, his instructions 
are complete. But unless the chart belongs to a standardized 
set in which the scale has been fixed, the draftsman will prob- 
ably determine upon his y-axis scale after a study of the data 
itself. In this case, he enters the scale in hand-lettering and 
proceeds with the plotting. If this scale also is to be type- 
written, however, he would enter the scale-figures only in 
pencil and the typist would enter the figures permanently 
last of all. The considerations affecting the choice of scale 
will be found in the following chapter. 


CuHaPter XVII 
SCALES 


Technicians are fond of describing a scale in puzzling and 
abstruse language. Yet, as often happens, the thing itself is 
so simple that a child can understand it. It is usually defined 
as a ratio—the ratio between actual distances in the space 
charted and equivalent distances on the chart. This ratio of 
reduction or enlargement is important to engineers but not 
to the maker of mathematical charts. We therefore use the 
word scale for the calibrations measuring distances on the 
chart. To linear distances, both horizontal and vertical, we 
assign arbitrary values and the figures which tell us these 
assigned values form the scale. 

Curve charts take up two dimensions on the paper, that is, 
they have both a vertical and a horizontal axis, and therefore 
require two scales. These two scales may or may not be alike. 
When they are alike, we have what might be called a normal 
projection. Imagine a simple chart, the field of which is square 
and the two scales of which are alike. Draw a straight line 
from the point of origin or lower left hand corner of the chart- 


168 CHARTS AND GRAPHS 


field at an angle of 45° to the horizontal and extend it diagon- 


ally across the field to the upper right hand corner. This line 
passes through all points having equal co-ordinates, that is 
equal values along both axes. Now let us see what happens 
to this line when one or the other scale of the chart is changed. 

Suppose we shorten the vertical scale to half of its distance. 
Relatively speaking, this is the same thing as doubling the 
horizontal scale. (By half or double the length of the scale, 
we mean assigning the measurement values to distances half 
or double as great.) Now the result of this will be very 
noticeable upon the slope of the straight diagonal line passing 


eae eat 
=e ' zi] = 
as es wae 
= i 
Fig. 148. 


through points having the same values as before, for the line 
will rise only half as much as before. If our line were a curve 
wiggling across the chart, its wiggles would be half way 


Fig. 149. 


flattened out, giving us the impression of much less fluctuation 
than formerly. But as a matter of fact it is exactly the same 
curve as before, only its field has been changed so as to dim- 
inish the vertical oscillations or wiggles. 


— 


SCALES 166 


Fig. 150. 


On the other hand, suppose that we increase the vertical 
scale to twice the length of the horizontal scale. This is the 
same thing, relatively, as reducing the horizontal scale to half- 


Fig. 151. 


size. Now see what happens to the diagonal line. Its slope 
becomes far steeper than originally, as it must climb to twice 
the height in the same horizontal distance. If that line had 
been a curve, snaking its way across the paper, its wiggles 


CHARTS AND GRAPHS 


would have been twice as great as formerly. It would have — 


- given us the impression of a very unsteady and changeable 


Fig. 152. 


proposition indeed. First way up, and then way down. Very 
hard to tell just where it is going to go next. Unstable, un- 
reliable, fickle—these are the conclusions we should have 
formed of the items that were charted, arid yet those items are 
precisely the same as appeared on the second chart above 
described where their movements appeared to be very even 
and regular. 

In short, the scales on which a curve is drawn can affect 
very much our impressions of the data by magnifying or minim- 
izing the apparent movements of the curve itself. Of course, 
this does not mean that the relative height from the base-line 
of the various points on the curve have been altered. If you 
have been careful to show the base-line always, the base-line 


SCALES | 171 


itself will approach nearer to the curve as the vertical scale is 
reduced and the wiggles are flattened out, and will recede 


ie 
Biba 
Fig. 153. 


farther from the curve as the vertical scale is enlarged and the 
wiggles are exaggerated. But it means that the oscillation or 
fluctuation of the curve will have been made to appear more 
violent or milder according as either of the scales is changed. 
And it therefore behooves us to give serious thought to the 
matter of scales before’ we determine upon them finally for 
any particular chart. As a matter of fact, we may have to 
try out several combinations of scales before we find one which 
gives just the right amount of emphasis to curve fluctuations 
to suit us. 

‘=. Now where our chart size is unlimited,. and we are free to 
extend the scale and field in either direction as far as we wish, 
this rule of try, try, try again might be perfectly feasible. 


172 CHARTS AND GRAPHS 
Perhaps, in that case, we would generally come back to the 
normal projection or combination of two similar scales. If, 
as often happens, the scales measure different and incomparable 
(technically “incommensurable’’) quantities, such as years on 
one axis‘and dollars of sales on the other, or length in inches on 
one axis and weight in pounds on the other, we could change 
these to percentages (each of its own total or maximum), and 
consider the percentages commensurable. 

But generally, the space available for a chart is limited. 
If it is to appear in a book, the size of the book-page must be 
conformed to. If it is one of a set of charts, a uniform chart- 
size increases the attractiveness, if not also the simplicity, 
both of the set and of the individual chart. Even if the chart 
is to appear entirely alone, there is much benefit in avoiding 
unhandy sizes. Moreover, worrying through a succession of 
trials consumes time and energy, a needless waste if it is true 
that we can determine beforehand merely from the data 
itself what will be a satisfactory combination of scales. 

Let us consider the horizontal scale first. In the previous 
chapter we have already found certain considerations which 
will affect the arrangement of the figures for this scale. These 
figures will be placed immediately below the base-line or bot- 
tom of the chart. Normally, they will be written on edge, up- 
ward, or typewritten after the paper has been fed into the 
typewriting machine sideways. The ordinates or-vertical lines 
to which the figures belong will then, if extended down the 
chart, pass through the figures, cutting across the middle of 
each digit. In the same way, above the top of the chart the 
data-figures will be placed, each on line with its own scale- 
figure and plotting point and each so placed as to be similarly 
cut by the extension of its ordinate. 

It takes no brains to see, therefore, that the horizontal 
scale must be large enough to permit entering the figures, no 
matter how condensed, of the data. As a general rule, type- 
writer intervals, heh are in picas or sixths of an inch, are 
about as small as your horizontal unit-distances should, be.l 
And if you have a short series of data, you can double or 
treble this distance without expanding your chart too much. 
In fact, the curve is more easily read when the horizontal 


1 All typewriters can be especially equipped, at a slight extra cost, with any 
desired interlinear distance and there is one machine, the Hammond, frequently used 
in academic work, which has intervals of one-ninth instead of one-sixth of an inch. 


SCALES TR 


units are about typewriter double-spacing distance apart, that 
is, three to the inch. — 

We are assuming here that your finished chart is to occupy 
a sheet of paper about standard letter-size, 81% by ll inches. If 
larger sheets are to be used, you will modify all dimensions ac- 
cordingly. Where charts are to be exhibited in a large room 
to a large audience, they must be many times larger and all 


eee ees ii Ore OR ee ee eae be 
8OGgaad s3a 70 COcoaagn aad dS oooo 
AXeseaaxn 2 aeaaosaann kaw aw Net nOazA 
S Q he $s p> (=) cq a0 Qu, » > ro) 

Q, @ S 5) 3 © ca) (e) @ 
S a g <x a OS) ue << uw (@) a] Q 
S Q be $4 Db (= qo ce) 
a ® Cy Qu. o S 3 A 
bar) al x aA a ca\ < 

ba be bs 

3 ° < A. 5 g 
ar) <7} a < P| Le) 
& He) he t > 

® 3 Qu Gu 
5 ce 3 < 3 
S Q bh bu 

® Gs Qu 
5 Gy P| <x 


Fig. 154. Examples of Convenient Horizontal Scales. 
Facsimile Typewriting. 


lines and lettering correspondingly heavier. The most excel- 
lent chart in the world is virtually useless to the man who 
cannot see it, and you must not forget the distance from 
which the chart is to be viewed. For ordinary study, however, 
as well as for convenience in handling and in filing, the 814 by 
11 basis is satisfactory. It can always be enlarged by photo- 


stats, or photographed on lantern slides for very large pro- 
jection. Ae 4 
The range of this horizontal scale then depends largely on 
the number of items in the series, to be plotted. A series 
which contains more than thirty items had best be cut up into 


Fig. 155. Showing One Month by Days on Letter-size Paper. 
Single-spaced for typewriting data. : 


two charts, each one of which will run across the shorter dis- 
tance of sheets of the 814 by 11 paper. You can generally do 
this by breaking up the series into convenient segments. Thus 
if the data is monthly, break it up into years and present a 
year on a page. If the data is annual, break it up into ten 
or twenty year groups. Where it seems inadvisable to break 
up the series into parts this way, double width sheets can be 


SCALES 175 


used, either folding up into regular size or not, as desired. If 
you wish to run the chart along the long distance of the 814 by 
11 paper, the space for attaching data above the chart will be 
much restricted, but if the data is limited to one or two col- 
umns, this is no disadvantage, and you can get as many as 


ec te 
Fig. 156. One Year by Months on Letter-size Paper. 


Double-spaced typewriting. 


fifty-two items on a page, thus enabling you to show a year 


by weeks. #Y 

When you can do s0, it 1s always well to make the hori- 
zontal distances one-third inch each, or double typewriter- 
spaced. In this case, you cannot count on more than a dozen 
or fifteen units crosswise on the paper and twenty-five length- 
wise. The advantage of the wider spacing, as has been said, 


176 CHARTS AND GRAPHS 


lies in the greater ease with which it is read, neither its curve 
oscillations nor its data figures being so confusingly close to- 
gether as when smaller spacings are used. Moreover, the chart 
with fewer items on it will generally be more closely studied by 
the reader than one with a great mass of detail. In fact it some- 


1910 


Fig. 157. One Decade by Years. 
Triple-spaced typewriting. 


times pays to omit minor details in the data and make the 
items fewer and more important, in order to reduce a great 
amount of detail to a simple series. Thus the daily stock quo- 
tations would require a very large chart for their presentation 
during a year, while the weekly and sometimes the monthly 
average quotations will be just as significant, and far simpler, 
to the reader. 


SCALES si 


Now as to the vertical scale.2 The first general rule is that 
the highest plotted points on the curve should ordinarily reach 
about two-thirds of the way up the field of the chart. This 
gives the best results, because the top of the chart neither 


Fig. 158. One Quarter-century by Years 
Single-spaced typewriting. 


crowds the curve too closely nor does the space above the 
curve seem to the reader unnecessarily large. If the top of the 
chart is too close to the curve at any point, the reader may be 


2It is to be understood in the following discussion that what applies to the posi- 
tioning of a single scale for a single curve applies also to two or more scales for two 
or more curves when these are shown on one chart. Unless there is special reason for 
having one curve below the other, or for using a common scale for both curves, the 
second curve may have its own scale (lettered on the chart beside the first scale) 
specially positioned, like the first scale, to bring the second curve to similar heights 


upon the chart. 


*aedeg eZIs-19}307] e]GnN0q uo syeeM Aq aeaxouQ “ESI *3IY 


| 3 *PlOF PY2 SMoys suT] BUC] ay, *suTITIMad Ay peceds-ai3urg 
: 


CHARTS AND GRAPHS 


178 


led to measure with his eye distances on the chart from the 
curve to the top line, instead of from the bottom line. 
When a single long series of items is to be carried through 
many charts, one after the other, forming a set in which the 
individual charts show only parts of the series of data, it is 
important to have the vertical (as well as the horizontal) scales 
uniform throughout. The uniform scales are necessary that 


Fig. 160. One Year by Weeks on Letter-size Paper. 
Single-spaced typewriting. 


the charts may be individually compared, or “fanned out” into 
one long series of continuous charts. And if the scale be such as 
to place the highest point in the whole series three-quarters of 
the way up the page, in one chart, there may be other charts 
in the series, in which the curve will hardly leave the zero, or 
base-line. This cannot be helped without enlarging the scale 
for these smaller parts, and so destroying the comparability of 
the charts. There is no help for the low charts in this case, 
nor is help really desirable, since the lowness of the curve at 
certain points is the significant fact to be shown. 

The size of the vertical scale depends therefore upon the 
amount of the largest figure in the date. We must glance 


° 
000‘000' 00% 
000°000' 009 
000‘000'000" ® 
I t+ 000'000'00s'® 
: 000°000' 0003 
N 000'000'00S*& 
Re = eS 
é se Sata) hero pe ate 
a Se Se eK oe an Ng) ol ws 6S i AD ea? ae ae Tr ear ~ 
: poe eee eee esas canine cee ae son wl st Bos SE. Ss § 8 Bes e Se teaden 
cee a SSIS  ANEmE Ad Go ho oe 0) Semi Oo @ 9 o)| wo. 8 @ 3 a BY wi wo = & 2 
S CUCM SIIB Ee US IES) Soe OR Oe Nae Se} nO 8 o's & SSS Se “s4srteo 
eee reeee oe 8B 2/ 8B BRB 2ees = 2 = eee e% 
Be oeeeeee ten 62 tree 6S sja oe 6 8 © 85 6) 8 38°38 5 3 88 8 8 8 
c. 
- 
(220Wa2D Fo {eusros R19, AON - @24N0S) 
(eres yieiseg Tes1d413g) seibiS pestUup 
1 J02 10 COO*OOTE Ie Pe Bitsep 
@uUOT esODJOOUT mew JO JELTSeD Ses TueIIO 
. 0261 @uoyye20d100u] aon 616) SUOTSIOTIOOU] MON ele) SNOUILE2ISEOONT EN 


180 


a 


SGALES 181 


through the columns of data to be charted and observe the 
highest quantity in the series. Of course if this is a freak 
quantity, we can disregard it and select the next highest quan- 
tity (leaving the highest one to extend clear out of the chart 
if it will). Having determined on what we shall consider the 
high point or “peak” of the data, let us substitute for this a 
round figure, which we shall position about two-thirds or 
three-quarters of the way up from the bottom of the chart. 


FIRE LOSSES 
United States 
1876-1920 
(Source:- Journal of Commerce) 


CO 


£282333338 388888888 gb888 
UE CELECELECECEEEETELCEEEECEEEEETEEEEEECEEEEEEEE 
$ ¢ 


sa 


300,000,000 =e o i | | 


= 


a 
wren ead eredee cea oo 


200,000,000 t 


100,000,000 h 
ha || iil || | 
Al fl 


Thus if the high-point or peak is $8,370,000, let us take $8,000,- 
000 as the scale-making figure. Now if this peak is approached 
by several others, that is, is no unusual value, let us make it 
slightly lower down on the chart, but if it is unusual, and the 
general level is nearer six or four million, let us make it slightly 
higher up. Anything between a third and a quarter of the 
distance below the top of the chart is sufficient. Assuming 
that the series contains several figures near eight million, we will 
select the slightly lower position, and place eight million two- 


Se SM Ns eee ne See Om Yee nee ae ae 


182 CHARTS AND GRAPHS 


thirds of the way from the bottom of the chart to its top, In 
other words the entire vertical distance will be divided into 
twelfths, each representing one million dollars. In this way 
our vertical scale has been determined. 

Of course, the size of the chart itself has not yet been 
settled. Its width we disposed of under the head of horizontal 
scales. But so far we have not settled its total height. We 
have only decided the number of parts into which that total 


Jan 5,902,000 
Feb 5,640,000 
Mar 6,413,000 
Apr 6,494,000 
May 6,309,000 
Jun 6,186,000 
Jul 6,329,000 
Aug 6,261,000 
Sep 6,350,000 
Oct 6,819,000 
Nov 6,147,000 
Dec 8,370,000 


United Cigar Stores Co, Sales 
1921 
(Dollars) 


(Source: Survey of Current Business) 


Fig. 163. 


height will be divided. But the total height need give us 
little trouble. For a drawing on ordinary letter-size paper, 
the chart field, that is, the co-ordinate rulings, should not 
cover much more than half the height of the sheet of paper. 
There will be some space needed at the bottom for the hori- 
zontal scale figures, and considerable space should be left at 
the top for the data and for the title to the chart. So on a 
sheet of paper 11 inches high, the chart can best be made 
about six inches high. And in the example we have just con- 
sidered, where this total height of six inches will be divided 
into twelve parts, each representing a million dollars, it is 
easy to see that the ordinates or horizontal lines should be 


spaced half an inch apart. The scale ratio is } inch to $1,000,- 
000. 


SCALES 183 


There are many devices for dividing a length into desired 
divisions. The case of ten, fifteen, or eighteen or more divi- 
sions is no more difficult in a six-inch space than that of 
twelve divisions. Both engineers’ and architects’ rules divide 
the inch into various useful numbers of parts and when we 
desire an odd or fractional number of parts per inch, we can 


United Cigar Stores Co. Snlee 
292) 


(Dollars) 
(Source:- Survey of Current Business) 


S S 
fe} 
o 9 
‘= = 
nN fo} 
oO + 
n oO 
o wo 


6,418,000 
6,494,000 
6,309,000 
6,166,000 
6,329,000 
6,261,000 
6,350,000 
6,819,000 
6,147,000 
8,370,000 


10,000,000 


7 


a 
@,000,000/—} + ial sane 


A fh 
6 ,C00, 000 it i 
————- + +— | i fae! 
{ 4 
@,000,000 eae I ll I 
2,000,000 PAG 3 ~ 
1,000,000 $+ | i 
° ii a4 4 
» 
3 


be he > a 
s © s a @ EI 2 
> & = 


wo oD > 
3 o ° h 2 
3 uo5> 39> ££ ow 2 © 


Fig. 164. 


easily get them by drawing parallels from the corresponding 
divisions on a regular scale laid off so as to form a triangle 
with the desired scale and the last parallel. However, as we 
had considerable margin of choice in deciding the number of 
dividions, we can always find round figures which will work 
ily in the given chart field. ' 
eee. s that a standard “field” about 4 inches wide and 
6 inches high, has already been adopted by a great many chart- 


184 CHARTS AND GRAPHS 


e of 


gineer’s rule. 


50, 30 or 60, and 20 or 40 sides of the En 


The fourth is specially adapted to percentage data, requiring a scale from 0—100%. 


g Papers published by Mr. John Wenzel, Yonkers, N. Y. The first three ar 


Coniinercial Forms Available. 


the type described in the text to fit the 10 or 


Four Useful Chartin 


Fig. 165. 


2 poston ma wang 1 


‘ ’ 
SCALES Te 


makers, and by some publishers of chart-paper. This standard 
pepared chart-form is very useful. It is printed low upon 
regular letter-size 8!4-by-1l-inch sheet, leaving the necessary 
space below the chart for horizontal scale figures and a great 
deal of space above the chart for data and title. It is generally 
printed with the horizontal rulings only, so that any desired 
number of vertical rulings can be drawn in to suit your hori- 


SCALE 


Fig. 166. To Obtain a Scale Smaller Than Those Given by the Ruler. 


zontal scale. And the horizontal rulings are printed without 
any scale-figures for the y-axis, so that you can adopt whatever 
vertical scale you please. Three different rulings are made, in 
which the interval is either a fourth, a fifth, or a sixth of an 
inch. These three forms are sufficiently different to enable 
you to place a point almost anywhere you wish on the field, 
merely by selecting the right ruling and attaching to it the 
proper calibrations or scale-fgures.? 

3 A single chart-form, which has intervals of one inch up the paper between hori- 
zontals, can be conveniently used in place of the three, when only a few charts are 


to be made. It is a master-form which can, if desired, easily be converted into any 
of the three by ruling in the proper number of intermediate horizontals. 


Pi ae et Be ey ee RS re set Se ee pe ps Ce en ge 
} Ee 7 { J . ° " 


186 “ CHARTS AND GRAPHS™ 


Moreover, if you are doing much plotting, it will help you 
greatly to use what is called an “‘engineer’s scale”’ ruler, which 
can be obtained with the inch divided into fourths, fifths, and 


Courtesy of Keuffel & Esser, N.Y. 
Fig. 167. Engineers’ Triangular Rule. 


sixths, coinciding with the three types of prepared chart- 
paper, and into other even fractions, namely halves, thirds, and 
tenths of an inch. With these six scales suitable for use with 
the three types of ruled paper, you can conveniently draw up 
any curve in any position on the field. And if you have not 
access to the specially ruled paper and rulers, you can of 
course easily prepare them for yourself. 

For this chart field which measures 6 inches high (or for 
any other given size of chart) a general instructions-table can 
be used by which, without any figuring, you will know what 


SCALE 


Fig. 168. To Obtain a Scale Larger Than Those Given by the Ruler. 


ais 


i 


type of ruled paper, what edge of the ruler, and what values 
or calibrations in the chart-scale you must use. In the ac-. 
companying table for 6-inch high chart-fields, it is assumed 
that you will position the peak of the curve about two-thirds 
of the way up the chart-field. You have therefore only to 
glance through your data and find out the amount of the peak 
or largest quantity in the series and with this figure in mind, 
consult the table and find the round figure therein which is 
nearest it, and proceed as for that round figure. The only 


8 


w 
a 

a 
© 


AAAI TIAAIUATNAIANIIE 
AUAAAUAUARANARAAAAAUREAUEIBANANIAN 


@ o 
we oe 
wo oO Lo 
i) ~ 
n Ca 
ees 
vs 2 3 Oo @ r 
~~ n 
& 5 
n n 
> @o 
a a 
nr a 
a co 
cu ie] 


“8 A 4.0 
1.0 2.0 6 
6 oa 1.2 sii 2.4 es wae isa 
“5 .6 ep——s-12.0 1.2 1.6 2.0 2.4 3.0 4 
+ 1.0 2.0 2.5 
4 5 We 8 yee. 1.6 nl 
4 8 a6 2.0 ' 
aS 6 1.2 
3 4 a6 8 1.2 1.5 2 
4 .8 
é az 4 8 1.0 
a2 i 1.0 
1 1 22 ‘ : 4b— .5 5 
4 ol +2 2 2 2 = , 
05 j 1 a : . 5 
6 S 0 0 Fy 0 0 =o 0 
60 60 40 60 60 40 60 60 40 40 


Fig. 169. Examples of Convenient Vertical Scales. 


Reduced from Standard 6-inch Field. 

Shows scales with 10, 12, 16, 20, 24, 32, 40, 48, 60, 80 and 100 at the distance of 
two-thirds of the height of the six-inch chart-field, by the use of the three rulings 
for 40, 50, and 60 sides of the ruler. 


thing to remember is that the position of the decimal point 
does not matter. Your number may be .0003 or .3 or 3.0 or 
30 or 3,000,000; you will always find its plotting instructions 
under the first “‘significant”’ digit, namely, in these cases, the 
figure 3. ee 
These apparently arbitrary rules of thumb are justified 
only so long as they serve to produce the best results. Your 
real purpose is to show the data most clearly and simply, 
either to yourself or to someone else. The chart is a window, 


188 CHARTS AND GRAPHS 


RDINARY FIELD 6 INCHES HIGR FIELD 8 INCHES HIGH | FIELD 10 INCHES HIGH 
ORDI’ 


PEAK 
On 
IN Standard Fields 


ENTIRE 


SERIES 


7.35 - 7,50 


7.50 - 7.60 


Increase 60-field by 2/3 Inch 
2. 30 


TABLE OF SCALES FOR CHARTS 


Fig. 170. Table for Vertical Scales with Engineer’s Rules on 6, 8, and 
10 Inch Fields. 


as it were, through which the reader looks out upon an illu- 
minating picture of the facts he is considering. Through this 
window he sees, if you like, a chain of mountains, whose 
height tells him the values or quantities he is considering. 
That he may see them to the best advantage, the window must 
be low enough for him to see the base of the mountain-range 


SCALES 189 


and high enough for him to see at least some sky above the 
highest peak. In general, the best view of the mountains 
would show neither too much nor too little clear sky above. 
And if the window is crossed with a framework for small 
window-panes, he can further judge of heights by the criss- 
cross window-pane lines. Your curve is the silhouette of that 
mountain-range, your field the tiny window-pane outlines, and 
you, the chart-maker, must use your own judgment and ar- 
tistic sense to place the reader’s chair near or far, high or 
low, in front of that window, to give him the clearest view. 


Cuapter XVIII 
PLOTTING-POINTS 


A point can be defined or located by its co-ordinates. The 
co-ordinates of a point are the two mutually perpendicular 
lines which pass through it and at whose intersection the point 
is located. One of these lines is its abscissa, the other its or- 
dinate. Neither of them need appear upon the paper; that is, 
they may both be imaginary, and it is therefore sometimes 
difficult to chart or plot a point precisely, or when plotted, to 
read its co-ordinates exactly. 


Fig. 171. 


Here is a simple example. Suppose you have a chart- 
field on which each abscissa and ordinate represent a unit 
value, that is the abscissa and ordinate are numbered consecu- 


190 


PLOTTING-POINTS : IgI 


tively one, two, three, four and so forth. Now suppose that 
you were to plot on the field the point represented by the co- 
ordinates, “x, 314, y, 4%.” You will look in vain for an 
intersection of lines with these values, because the co-ordinates 
of half units have not been drawn on the paper. Nevertheless, 
you can imagine the two co-ordinates, one of them half way 
between the ordinates of “‘x, 3” and “‘x, 4,” the other half way 
between the abscissae of “‘y, 4” and y, 5.” And at the inter- 
section of these two imaginary co-ordinates you can plot the 
point. 

This question of the precise plotting-point comes up very 
often in charts showing time by weeks, months, or years along 
the horizontal scale. Suppose you are charting the monthly 
steel prices in 1920. Down at the bottom of the chart you 
will place the time-scale with the words January, February, 
March, and so on under the ends of the vertical lines of the 
chart. Consider this scale carefully. What does it mean? 
It means that each unit of horizontal distance has been taken 
to represent one month and that all twelve horizontal units 
taken together represent a year, the year 1920. Nowifa month 
were a single instant of time, it would be very simple. ; We 
would then plot the figures for each month or single instant of 
time on the particular vertical line which represented it. But 
as a matter of fact, a month is a long period of time with a 
great many different instants in it, all of which go to make up 
a single month, just as twelve months taken continuously 
make up a single year. In short, we are no longer dealing 
with single instants of time but with continuous periods of 
time. Yet on our chart the horizontal scale shows the number 
of single points representing these months. Something surely 
is wrong. Obviously we must find particular instants for 
points of time, to correspond with the points on the horizontal 
scale representing time. 

There are two ways of doing this. The more scientific and 
accurate way is for us to seize upon the particular instant or 
point of time between months and represent these points of 
time by the points on our horizontal scale. The origin of the 
x-axis, or zero point on the horizontal scale will then stand for 
the beginning of the month of January. The first point on 
the horizontal scale will indicate the end of the month of 
January and the beginning-of the month of February. The 
second point on the scale will represent the end of the month 


Bi ia ici 4 bate 


192 CHARTS AND GRAPHS 


of February and the beginning of the month of March, and so 
on. In this case we see that the months themselves are indi- 


Fig. 172. To Plot Anywhere Between Ordinates. 


cated on the scale, not by points, but by spaces between 
points. Thus if we wish to plot the figures for January as of 
the 15th of January, that is the middle of January, we will 
find a point midway between the zero and first upright lines, 
that is in the middle of the first space on the horizontal line. 
To plot a figure as of the end of the first week in January we 
will locate the point only a quarter of the way from the begin- 
ning of the horizontal scale to the first point on the scale, that 
is, a quarter of the way from the first of January to the end of 
January. And the scale itself, that is, the words “January,” 
“February,” ‘“March,” and so on, must be placed beneath the 
various spaces between lines, and not beneath the ends of the 
vertical lines themselves. This method enables us to distin- 
guish prices at the various parts of each month, and is in general 
thé more accurate method of scale calibration and point 
plotting. 

The other method, however, is more convenient both for 
chart-maker and chart-reader. Let us assume that each month 


PLOTTING-POINTS 193 


has been condensed into a single instant of time and that the 
upright line or point on the horizontal scale represents only 


sc 2 ? b Oo 
J 2 o °o oe 
2 & a °o = 


tb he > gs a a. 
3 a a § 3 £ & 
< =a 2 oa =< 


Fig. 173. To Plot Only upon Ordinates. 


this single instant of time. What particular point of time in 
the month is in general the fairest one for us to choose to 
represent the whole month? Obviously the middle of the 
month or the middle of the fifteenth day. And when we plot 
the prices for January we assume that those prices are the 
average prices for the entire month and that it is fair to show 
them as the prices for this point in the month, namely the 
fifteenth of the month. In this case our scale figures will be 
written immediately underneath the ends of the vertical lines, 
that is, the word “January” will appear under the first ordin- 
ate and not under the first open space, the word “February” 
will appear under the second ordinate, and so on. Clearly 
this method is not scientifically so accurate as the first method. 
But as it is much more convenient, it is the ordinary method 
of plotting a time series. When you use it you must remember 
the assumption upon which it is based, namely, that the entire 
period has been condensed into a single moment or instant of 
time and shown as of that moment. Whether that single 


194 CHARTS AND GRAPHS 


moment be at the middle of the period or at some other time 
during the period will depend upon your data, and somewhere 
about the chart a memorandum should be placed showing 
what particular moment in the period shown, the figures and 
plotted points represent. 

This second method can safely be used whenever the periods 
for which the data are charted, are uniform and equal periods 
of time. The method becomes very difficult and confusing 

i H i 


i FOOD PRICES IN FRANCE AND GREAT BRITAIN 
Index Numbers of Retail Food Prices in France, Great Britain, and United States 


(July 1914 = 100) 
(Source:+ Bureau of Labor Statistics) 


France (For current a 2 = 4 

(excluding Paris) Quarter) = " ” - 

Great Britain ' (For current ‘6 S3 2 eS: 2 a 
two Months) “ n ) n “ x 


United States (For current 
Month) 


500 


400 


300 


200 


100 


Fig. 174. Data with Different Intervals. 


however, as soon as the time interval of the data changes. 
Suppose that a part of the year was represented by monthly 


PLOTTING-POINTS | 195 


average figures and part of it by weekly*avetage figures, and 
perhaps also a part by quarterly average figures. You would 
have to watch your step in plotting these various periods by 
the second method. But by the first method it is all smooth 
sailing, for it is easy to plot different points in the space when 
the spaces represent the months. It is also easy to plot a 
single point in the middle of three successive spaces (as for a 
quarterly period) by the first method. The first method, as 
has been said, is sound and logical and should be used whenever 
you are in doubt as to the plotting point on the scale. This 
means also that it should be used whenever the time intervals 
are not uniform and regular. 

It is not necessary to say that one of the reasons why points 
should be correctly plotted is that the reader of the chart 
should be able to ascertain the values they represent directly 
from their co-ordinates on the chart. And the reader may 
have particular need for these values, not at the points plotted 
from the data, but at other points along the connecting lines 
which form the curve. The technical name for the process of 
locating points on a curve between. given ordinates is “‘inter- 
polation.” Just as we can interpolate for points on the scale 
when plotting given points, so also we can reverse the process 
and interpolate for the data of points upon a given curve. In 
the foregoing we have shown how to interpolate for plotting 
points in making the charts. Let us now consider the reverse 
process of interpolating for data, either in the making or in 
the reading of a chart. It is to this process that the term 
interpolation is ordinarily applied. 

Let us suppose that we have a chart compiled from data 
which is incomplete, that is to say, the months of June, July 
and August are missing. Our chart will show a curve extending 
over the first five months and the last four months of the year 
but there will be a gap during the three missing months. Yet 
we know that the phenomenon was in existence during these 
months. Therefore we cannot leave this gap vacant, for to 
do so would imply that the phenomenon had ceased to exist 
during that time. Now if we can make no guesses whatever 
as to the value which was missing, we should simply draw a 
dotted straight line across the gap connecting the two nearest 
known points of the curve—dotted or broken to indicate that 
data is missing and the curve for that distance is guess-work. 
If however, we can make a shrewd estimate as to the shape of 


196 CHARTS AND GRAPHS 


the curve across the gap, perhaps from a study of the same 
phenomenon in other years during the same months or from a 
study of similar phenomena during the same year, then we 
will not draw a straight line across the gap but will shape the 
cur ve over the gap in that way which we think most likely to 
be true. It would still be a good plan to use a dotted or broken 
line for these estimated or interpolated months. In any event, 
we have now assigned values to these months which were 
missing, by interpolation from a study of the surrounding 
ones which were known. The values of the interpolated points 
for these missing months can be read off from the chart and 
would form estimates for them with which we can even fill 
out the data record. 


TRADE UNION MEMBEXSHIP OF THE WOKLD 


Number of mambers in 20 Countries 
1910-1919 
(Source:- International Labor Office) 


S -eead ne 
S 3 3 3 = 8 ° e ° 
oO 3 oO . - « 
Number wo o Sy oO o S = S S 
A e ~ bid Nn x o oO fo) Oo 
ot oO nN i) n 2 a a me 
= « s = 
lembers “ o > n 
Ly We kee ee pe rg ee 
45,000,000 


40,000,000 
35,000,000 
30,000,000 
26,000,000 
20,000 ,000 
15,000,000 
10,000,000 

5,000,000 


Fig. 175. Interpolation for the Period of the War and Extrapolation 
for the Years after 1919. 


Interpolation for intermediate data from a completely 
known curve is also frequent. Thus if a curve shows the values 
at certain known points, we can easily secure the values at 
other in-between points by noting the points passed through 


ial 


PLOTTING-POINTS 197 


by the curve, that is, by interpolating for them. This guessing 
or estimating process can also be carried out beyond the limit 
of the curve to points lying outside of the range of the known 
data. A frequent example of this is the well-known process of 


O1t ConsuMPTION 


Millions of Barrels 


oe 19)2 1913 1914 1915 19l6 I917 1918 1919 1920 1925 1930 
From Joseph E. Pogue’s ‘‘Economics of Petroleum." 
Fig. 176. Extrapolation. 

Here it is the linear trend (having a mathematical formula) that is projected 


into the future. 
extending a curve into the future to predict or forecast what 
will happen at a given time to come. The curve is simply 
projected to points outside of rather than inside of the range 
of its data. This latter process is often called extrapolation. 
The processes of interpolation and extrapolation are capable 
of very general application. 


CHAPTER XIX 
COMPOSITE CURVES 


With the plain single-curve-chart, the reader of this book 
is now supposed to be thoroughly familiar. And as has been 
repeatedly indicated, the multiple curve-chart is formed by 
merely bringing together upon a single chart two or more 
curves, which may or may not cross each other. When two 


PRODUCTICN OF AUTOMOBILES 
Numher of Passenger Cars and Trucks Produced 
United States 
1913-1921 
(Source:+ Rational Automobile Charber of Cormerce) 


ce Oe ee ee eee ts 
. S88) # SiiGrsi Seas 
he Se ee Ee” ee 

ee ee tere, eee tere eet ae 
Eero) oat. mn PP ten 


400,000 2,060,000 


$00,000 1,600, 


200,000 1,000,000 


100,000 500,000 


20,000 100 ,000 


1917 
1916 
1919 
1920 


1921 


Fig. 177. Each Curve Has Its Own Vertical Scale. 
198 


COMPOSITE CURVES 199 


curves cross at small angles, it is the better practice to dis- 
tinguish them clearly, either by the use of different colors, or 
by the use of dotted or broken lines for one and full lines for 
the other. An older but slightly more confusing practice is to 
adorn one curve with small circles at its plotted points, another 
with small crosses, and distinguish other curves with double 
lines, wavy lines, lines with small cross-lines and sometimes 
lines of different thicknesses. In general, the best results are 
now secured by smooth lines, either colored or black, full or 
broken or dotted, but all of about equal thickness and visi- 
bility. Too many curves upon a chart are far worse than too 


; : INVENTION AND WAR 
Number of patents issued during the Civil War and during the World War 
i . United States ; 
{ 1850-70 and ‘'1903-20. i 
‘(Source: - U. S. Statistical) Abstract) 


o load N Lr ba w 0 & @o on o 

eA Ca eC) Re ONG OE TSS Bias o o 0 6 0 to © 

Civil 8 TN, ade reg SUNY AST SUA TE OM Sete gL Me © 5 
war py - 


Fig. 178. Each Curve Has Its Own Horizontal Scale. 


200 CHARTS AND GRAPHS 


few, and except for special laboratory work a chart should not 
normally carry more than three or four curves. The value of 
attaching data is so great that it is unwise to dispense with 
data, and yet if too many curves are used, the data will bulk 
up disportionately. That a few curves can be easily distin- 
guished without recourse to special adornments or thin and 


1915 1916 1917 1918 1919 1920 1921 


Fig. 179. 


Number of persons employed in industrial establishments in New York State 
and in the United States (figures for December 1914=100%)—Permission of 
Mr. Carl Snyder. 


thick lines, is obvious. The chief use for extra heavy or wide 
lines in curve-making should be for emphasis, as in the case 
of one curve for the average or total of the other curves. 

A thorough knowledge of the curve-chart, however, requires 
at least a passing acquaintance with its sisters and its cousins 
and its aunts. We shall therefore hold a reception and intro- 
duce the most important of these. Beforehand, however, let 
us whisper a word in your ear about them. They are, none of 
them, such all around good fellows as the plain curve-chart. 
They are not so flexible and universal in their uses. Each 
answers excellently to certain limited types of data. We shall 
try to make you acquainted with the particular style of data 
for which each is best suited, as we meet them. 


201 


COMPOSTTE CURVES 


Consider the case of daily stock quotations on the exchange. 
For any particular commodity, a dozen different prices may be 


*saAInZ jo pesysuy seuoZ “*OsST *S1.J 
“AN ‘ISUDUUY AY] UlOsT 
*"yreqd sit}? JO woyyeredosd 64} Ul Posn SHI0Is AIJIJ OW JO SUINIOA ApYa0M Bore SIjYAr OY} PUT SOTBS Jo- 
@UINOA ATH9OM [2}]0} SMOYS Lo1e HOVIG O43 UoTzod JeMo] oy} UT “S[fex VAyJ-A}UOM} IOJ SaunZjZ FZuypuodsetico oy} vase a3yy og pus ‘STBEIISNPUT 9AJJ-A]UOMI ay} Jo adIud OBVICAB ALE, 
WMO] PUT Js9q43T4 OU} AIPM YO JOJ EMOYS Lore AIA CML “SPVOATTEL JLeY PUG s[efrIsNPUF JLeY ‘SHO 43313 JO CoJAd eFEIOAG Suysojo ayy SMoYS eUTT YOLIG BM) UONIOd seddm guy UT 


23 ak 
: 
E 35 fe) 
re : aa aS ce 
cr aq AON, WO, Jag any, Ajng oung¢Aeyy, Ady Jey qay wef js9q AON 499 Jdagsn Aine ounrAeyy idy iey qq wer. . 
os F 0G 
ss see cS 
09 : 09 
$9— : 9 
OL = OL 
GL Seer SEE SL 
08 — 3 8 
S8 c8 
06 E = 06 
66 3 : z 6 
= : + 
00T: et see 1 00 
S0I-fe - sO 
Oe : Orr 
{SIE Sega ia a 
oe 2z6r 1261 % 


It is therefore customary to quote 


”? as well as opening and closing quotations 


e day. 


quoted in the sam 
“highs” and “lows 


202 CHARTS AND GRAPHS 


in the stock market record. Now if we plot upon the ordinate 
for each day both the high and low quotations, and connect 
the low quotation points to make a curve for low quotations 
and similarly connect the high quotation points so as to make 
a curve for high quotations, we shall have two curves illustrat- 
ing the same phenomenon, namely stock prices. The two 
curves show merely the extreme fluctuations of this phe- 
nomenon and the reader of the chart must understand that 
prices have ranged between these two curves. To make this 
situation obvious, let us shade the area between the two curves 
so as to make a zone. The shading can be done either with 
gray, with colors, with cross-hatched lines or with solid black 
or white, the co-ordinates being wiped out in the last case. 
This device conveys at once to the reader of the chart the idea 
that prices were not set at one figure alone, but varied con-- 
siderably within the same day or period of time. The device 
can be used for any case of data covering maxima and minima 


a 
— 
8 
east 
PO 
| | 
| 
— 


fe] 
1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 


Fig. 181. An Excellent Form of Zone-Curve. 


High, low, and average interest rates on commercial paper each year from 1831 
to 1920.—Permission of Mr. Carl Snyder. 


RETAIL POCD PRICES ) 
Teten Tumbers of tne Bureau of Labor Otatiotion SS 
Onited States 
Jez. 1919-Dee. 1921 
(Average 1913 © 100K) 


@un Pod Mar Apy Way Jun Jul ang Sep Oot Bor Deo Jen ob Mar Apr Way Jun Jul Aug Bop Cot Wow Deo Jan Pob Mar Apr May Jun Jul Aag Sep Uot Bor Deo 


All artlelee 186 172 178 162 188 184 190 192 168 189 192 197 201 200 200 211 216 219 219 207 203 198 193 178 172 168 166 162 145 144 148 158 183 163 162 


180 


162 162 165 172 176 170 171 166 161 157 166 164 159 160 161 170 171 162 192 188 165 177 171 166 169 161 154 167 156 188 168 157 163 147 
176 X74 177 182 187 181 185 177 170 165 162 161 166 167 168 179 179 191 202 196 193 188 178 160 163 165 167 160 160 160 161 160 164 148 


Rid roset 166 165 169 176 178 171 169 164 168 165 153 163 159 159 161 169 169 176 181 176 175 168 166 182 167 148 152 164 165-151 148 147 144 139 
Chuck roast 176 174 176 184 186 176 178 166 168 153 161 162 168 167 187 166 166 174 179 172 170 162 168 146 148 138 161 140 156 136 129 130 126 124 
Plate beef 161 161 16S 187 181 174 168 160 160 145 143 143 152 162 160 167 165 167 168 164 152 147 146 136 140 129 130 127 124 117 109 112-210 loo 
Pork vhope 193 180 184 197 206 202 220 223 219 211 200 18] 178 18D 186 206 202 194 208 219 238 238 210 167 171 156 168 177 167 162 163 181 179 172 
Baoon 217 206 205 212 210 212 216 214 206 196 169 186 186 186 186 191 196 200 208 203 202 202 196 276 171 166 166 164 161 165 180 162 169 153 


199 195 191 197 205 2Q6 211 212 205 196 188 186 187 188 190 199 206 216 222 224 224 222 212 186 160 170 161 16$ 161 182 190 197 191 180 
211 208 212 225 246 264 266 266 242 228 251 221 216 204 192 191 160 185 164 177 177 186 163 162 141 131 124 116 108 103 108 118 113 109 
188 186 193 202 204 200 197 196 194 189 184 164 197 210 216 224 221 216 211 212 214 207 20} 189 200 201 203 202 194 181 162 163,179 175 
218 147 140 143 154 155 164 174 163 209 235 261 240 199 161 163 163 156 166 164 206 234 260 266 229 139 121 99 97 101 122 138 146 171 
184 149 174 186 177 168 164 167 172 186 197 204 194 190 196 199 187 176 177 175 179 180 181 162 169 148 150 145 111 108 122 134 152 139 
201 186 185 190 191 192 195 197 196 192 196 196 196 196 194 194 194 189 186 183 184 164 140 176 175 171 176 169 143 153 183 146 148 149 
176 174 168 169 157 169 169 174 176 180 184 188 187 188 187 183 162 182 168 191 193 194 194 189 165 178 171 167 162 160 167 161, 158 160 151 158 
176 178 176 176 176 177 179 180 180 180 182 162 195 196 200 200 205 211 213 213 215 211 207 193 193 189 188 166 177 175 173 173 171 170 186 16S 
200 BOS 206 218 227 227 227 224 221 221 224 233 245.245 242 245 264 267 264 266 252 256 221 200 208 197 194 179 175 170 176 175 170 164 165 152 
207 200 197 200 207 210 217 220 223 220 220 220 220 217 217 217 223 260 255 280 227 213 197 163 173 167 180 153 150 160 147 160 147 143 140 137 
159 164 164 164 154 159 168 178 190 199 202 203 208 210 211 214 215 218 214 210 202 185 165 182 187 121 113 106 101 101 100 101 108 107 108 107 
188 182 171 182 194 224 282 294 253 224 229 253 $18 353 400 536 566 606 624 294 229 200 194 186176 163 147 136 129 169 200 247 235 206 188 182 
196 196 193 195 195 193 198 202 200 207 227 264 324 342 340 367 462 485 462 416 S93 263 236 191 176 162 176 176 158 142 129 156 133 126 122 118 
117 123 126 129 136 143 186 160 164 169 164 164 165 166 166 165 165 165 166 162 15S 146 139 133 129 126 125 123 121 120 120 119 119 119 119 lie 
127 126 129 128 128 129 130 130 130 131 131 127 152 132 135 136 136 136 157 187 137 133 135 133 133 131 131 129 129 126 127 127 127 127 127 126 


+ = 
I : f al 
| 
4 i a 
joe 
Lela pe 
| 
ela IE 
| | Le jig 
| {eral a 
(aa ae L 
abo — = + | 
— | | + 
| f ies + 
{ 
L ee | 
i +. 
1 i | | 
400 5 
| = 
15 eat | if | 
| 
He i 
4 je 
580 ale ; i ai = 
+} + i 
ees 7 5 
et 
L = 
300 
| | 
| 
260 + \ | 
200 
160 o . > N 
= 
i \ 
100 
}— = 
| ; 
60 | [ | 
|» 
[ 1 Oe 
| 
° f 4 Sep Oct Bow Lec Jan Fob Mar Apr May Jun Jul Aug-Sep Oct Now Dew 
dan Feb Mar Apr May ae Aug Sep Oct Now Dec Jan Fob war Apr May Pra Aug Sep low Lec Jan un 


Fig. 182. It is Useless to Show All the Individual Curves. 


404 CHARTS AND GRAPHS 


for a single phenomenon. Climatic conditions, such as 
humidity at morning and night, or temperature at mid-day, 
midnight, and noon, or tidal variations, or business statistics 
such as the margins of profit from individual sales, and in 
general all data having a considerable range of variation for 
one and the same thing, at approximately one and the same 
time, can be shown by this method. This type of curve-chart 
is commonly called the zone-curve. 

In a sense, the zone-curve is merely a short-cut for a large 
number of curves superimposed upon each other. In some 
cases you will have such distinct data that you could have 
prepared a large number of separate curves. If you put all 
these curves together upon a single chart, you will produce 
much the same visual result, so far as the reader is concerned, 
as you produce by means of the zone-curve, in which you 
merely plot maxima and minima and shade the space between 
them, Where the individual curves must be kept distinct 


B earsaey ee TTT 
za] [im a TT Te TT 
i A aa i | 
ais 


— 
Zz 
uy) 
5) 


eeeeeee | oe : 
a Att per : fi dd — 


4 ie S: 
ny 
frail li 
1915 1916 1917 


PL 
Permission of Standard Statistics Co. 


Fig. 183. An Excellent Adaptation of the Zone-Curve. 


however, and compared with each other, of course the zone- 
curve is of no use. The zone-curve, therefore, is not a sub- 
stitute for the multiple-curve chart. 

We have said that the connecting lines between the plotted 
points which form a curve, imply a connection between the 
items of the data. We have said that this connection is es- 
tablished by the variable nature of the stubs, or x-axis scale. 
It is now time to let you into the secret that these plotted 
points do not need to be connected. The “gun-shot” chart is 
an example of a curve without curves. It consists entirely of 
plotted points. It is useful for cases of data secured by sep- 
arate and often contradictory observations. Each observation 


COMPOSITE CURVES 205 


is plotted as a point but no connecting line can be drawn to 
other points and there is merely a large group of dots or plotted 
points extending across the chart and showing the result of 
various observations. This chart and its data differ from the 
zone-curve and its data in that there is no maximum or mini- 
mum known. Isolated points or dots on the chart may occur 
far outside of the general run or trend or zone of the main 
body of observations, such cases being due to freaks. errors, 
or other causes. 

Gun-shot charts are essentially a research device. They 
are often intermediate steps between the first data gathered 


Kw. _Kw. 
RPM RPM 
5 


Rw. 
RPA 


a 
Bet] 


900: Fearn aXiar 
pee eae 
ZAR ge 


200 


Kw. Kw Kw. 
RPM RPA R.PM 


From Leonard A. Doggett’s ‘‘Cost per pound of Electrical Machinery,” in the Electrical World, 


is Fig. 184. A Gun-shot Chart. 

and the final data reported. If we find after making a gun- 
shot chart of any particular observation, that all the points 
lie within a very narrow zone across the chart, we will be 
tempted to draw a line through this zone and so “fit a curve” 
to the plotted points. This is a sort of deductive reasoning 
by which we may often reduce a mass of data to a simple curve. 

A popular form of the curve chart is the “staircase” curve, 
sometimes erroneously called a “histogram.” The staircase 
curve is a direct throwback to the pipe-organ or vertical-bar 


206 CHARTS AND GRAPHS 


MAGAZINE ADVERTISING 
Number of Agate Lines of Advertising 
in Leading Magazines 
United States 
1913-1921 
(Source:- Printers’ Ink) 


14,688,000 
13,932,000 
13,764,000 
16,980,000 
17,880,000 
16,128,000 
22,680,000 


8 °> 
° 

°o o 
- 

oO 7 
ao a 
te) in 
. : 
i 

a «a 


30,000,000 


26,000,000 


20,000,000 


16 ,000, 000 


10,000,000 


6,000,000 


Fig. 185. The Staircase Curve is Almost a Bar-chart. 


1913 
1914 
1915 
1916 
1917 
1918 
1919 
1920 
1921 


chart. If you will recall the original definition of a curve given 
in a previous chapter, you will remember that a curve may be 
defined as a line connecting the upper end of the bars in a 
vertical-bar chart. Now if you will make the bars wide enough 
so that they actually touch each other and will then draw the 
outline or silhouette of the upper ends, you will have a curve 
made of rectilinear lines always parallel to one or the other 
axes of the chart. This is the staircase curve. Whereas the 
ordinary curve represents the end of the bars by mere points 
and connects these points with straight lines, the staircase 
curve gives full value to the entire width of the bar. It is 
the precise silhouette formed by the bar-chart when the bars 


COMPOSITE CURVES 207 


AMERICAN 
EXPEDITIONARY 
FORCES 


Ge 


424 


5. 
480 
320 2 if 
325) 224 \ 
176 
198 | 22° is3 


BOO R90 39 $00 $51 692 348 00 1169 1325 1462 1639 1796 1953 2112 2380 2658 3001 3437 363¢ 3623 3000 26242373 20581754 1308 936 STS 
APR.MAY JUN. JUL AUS SEPTOGT Nov DEC JAN FEB MAR APR MAY JUN JUL AUS SEPT OCT NOV DEC JAN FEB MAR APR MAY UN JUL AUG 
eb BAIL IEA Dt rs nO ALN a ae AL Acc sete EE MAU) A OIE A LARS et AA Beale Le EC a 


197 1918 mityt) 
Fig. 186. An Absolute Compound Pipe-Organ Bar-chart or an 
Absolute Stair-cased Band-chart. , 
Thousands of soldiers in the American Expeditionary Force on the first of each 
month.—Permission of Mr. Leonard Ayres. 


are packed close enough to come in contact with each other. 
As compared with it, an ordinary curve, directly connecting 
the midpoints of the: ends of the bars, is called a smoothed 
curve.! 

In some cases, the staircase chart is more accurate than 
the smoothed curve and its representation of areas lying be- 
tween the base line and the curve, more accurate. A little 
study will show you that the connected-line curve has cut off 
little triangles from every bar whenever the curve descended 


1 Beside the staircased (or rectilinear) and the smoothed (or line-and-angle) curves, 
there is still a third which is of such doubtful value and great hazard as not to be 
mentioned here. It is the rounded curve, in which no straight-lines or angles occur, 
but all parts of the curve are rounded off by means of “French curves” or by free- 
hand drawing. It is discussed in the chapter on Frequency Curves. 


ee CHARTS AND GRAPHS 


and added little triangles to every bar when the curve as- 
cended. These little triangles are sufficient to change the 


MAGAZINE ADVERTISING 
Number of Agate Lines of Advertising 
in Leading Magazines 
United States 
1913-1921 
(Source:- Printers’ Ink) 


13,764,000 
16,980,000 
17,880,000 
16,128,000 
22,680,000 
27,588,000 
17,592,000 


° ° 
oO ° 
° ° 
= «~ 

2 nu 
a fo) 
o a 
- « 

oe 

a oa 


20,000,000 


15 ,000, 000 


9 
191 
1917 
1918 
1919 
1920 
1921 


Fig. 187. The Smoothed and Staircase Curves Differ in Outline and 
Areas. 


area lying between the curve and the base line, and bounded 
by the two ordinates about the plotted point, and when it is 
important that this area should be accurately shown, you can 
not use an ordinary curve but must use a staircase curve. At 
other times the staircase curve is less accurate than the 
smoothed one, for its abrupt changes of level give an impression 
of abrupt fluctuations in the phenomenon charted, which may 
be wholly unwarranted. The considerations governing the 


COMPOSITE CURVES 209 


comparative value of the smoothed and staircase form of curve 
are treated fully in a later chapter.2 

The staircase curve is a popular form because it conveys 
at once to the average reader the impression of actual quan- 
tities between the base line and the curve. Readers who are 
confused by ordinary curves find less difficulty in under- 
standing this chart. It is not, however, so useful as the ordi- 
nary curve because a number of these staircase curves cannot 
be satisfactorily put together upon a single chart. Their 
vertical portions will so often coincide that it is hard to dis- 
tinguish them. The most that can be accomplished in the way 
of combining staircase curves is to put two or three of them 
together and use dotted, broken, and full lines to distinguish 
those which intersect. 

There is a certain type of data for which the plain ordinary 
curve closely imitates the staircase curve in its rectangular out- 
line. This is the case of data in which the values remain ab- 


Sx Na (a ae See, 
COMMON LABOR WAGES FOR 10 HOURS OF WORK 
UNITED STATES STEEL CORPORATION 


1916 1917 1918 1919 
Fig. 188. Pseudo-Staircased Curve. 


Note that so long as wages remain unchanged the curve must be a straight hori- 
zontal line and that when wages change the curve must be a straight vertical line. 
Hence the rectilinear form though truly a smoothed curve-—Permission of Mr. 
Leonard Ayres. 


1921 1922 


solutely fixed over a given period, and change only suddenly 
and abruptly. An example of this type of data would be the 
retail price of a single commodity, which after remaining at 


2 Cf. Chapter on Frequency Curves. 
1 Ofmiien 293" prose 


210 CHARTS AND GRAPHS 


seventy-five cents for a long period of time suddenly and on a 
single day jumps up to one dollar, to remain there for another 
long period. Obviously to plot the 75¢-value by a dot at the 
beginning of that period and the $1.00-value by another dot 
at the time of change and connect the two by a direct line 
would give the impression of a gradual change extending over 


ivan 
ae 


CERTIFICATES of LIBERTY 
INDEBTEDNESS ol BONDS 
192] 1921 
Fig. 189. 


Open market interest rates at New York compared with the discount rates of 
the Federal Reserve Bank of New York. Open market rates shown are for 
prime 4 to 6 months commercial paper, prime 90-day banker’s acceptances, 
certificates maturing in 4 to 6 months, and an average of the yields of 4 issues of 
Liberty Bonds and Victory Notes most frequently offered as security for advances. 
—Permission of Mr. Carl Snyder. 


the entire period. It is therefore necessary that this curve 


should be perfectly level until the change takes place and then 
jump up to the higher level and remain there. The curve will 
then have a rectangular outline similar to that of the stair- 
case curve, but the length of time or the length of the curve at 
any particular level is not regular and fixed. It is merely an 
accident that this picture has resulted in a curve with recti- 
linear outlines. It is not the same as the staircase chart. 


COMPOSITE CURVES De 


JAN. FEB. MAR. APR. MAY JUNE 
Fig. 190. A Pseudo-Staircased Curve. 


Call loan renewal rate and prevailing rate on prime 90-day banker’s acceptances 
at New York.—Permission of Mr. Carl Snyder. 


20 


BOND SALES IN HUNDREDS OF MILLIONS OF DOLLARS 


DURING FIRST HALF OF EACH YEAR SINCE 1899 
15 
Years cross hatched are those of tusiness depression. 
Note inoreased bond sales after each such period. 


10 


“Ye Uj 
Ze ZZ 
6) 
Permission of Mr. Leonard Ayres. 
Fig. 191. An Interesting Use of Shadings in a Band Chart, or Vertical 


Bar-Chart, 


212 | CHARTS AND GRAPHS 


A gay and giddy member of the chart family is the ‘“‘band- 
chart.” Take up any of the ordinary curves which you have 
made and with a soft pencil shade the entire area under the 
curve. This vividly reminds the reader that the data is rep- 
resented by the distance between the base line and the curve 
and not by the distance above the curve to the top of the 
chart, for it draws his attention forcibly to the lower part of 
the chart lying under the curve. You will remember the 


THE FAMILY BUDGET 
Divided as to Classes of Commodities 
United States 
1914-1921 
(Figures as of December each year) 
(Source:- Monthly Labor Keview} 


Total 103.0 105.2 118.3 142.4 174.4 199.3 200.4 174.3 
Miscellaneous 21.9 22.9 24.1 29.9 35.1 40.4 44.3 44.1 
Furniture and : 
Furnishings 5.3 S.c 6.5 lon! 10.8 13.4 14.5 11.1 
Fuel and Light 5.0 5.3 5.7? 6.6 7.8 8.3 10.4 9.6 
Housing 13.4 13.6 13.7 13.4 14.6 16.8 20.2 21.6 
Clothing 16.8 17.4 19.9 24.8 34.1 44.6 43.9 30.6 
Food 40.1 40.1 48.1 59.9 71.4 TSe1 68.0 57.3 
ys ZP 
AUDA: 
LEIBA Rt ce CK) 
Pret as = 
b Py | I 
LZ (fie i, 
ee ee 
a Sal 
| F000 | 
1914 1915 1916 1917 1918 1919 1920 1921 
Dec Dec Dec Dec Dec Dec Dec Dec 
Fig. 192, 


The Curves are True Only for Cumulations of the Layers. 


literal representation of quantities used in the bar-chart. And 


in fact the band-chart showing quantities by its shaded area 


COMPOSITE CURVES 213 


can be made in stepping form like the staircase chart as well 
as smoothed like the ordinary chart. The staircase band-chart 
is therefore even more of a throwback to the vertical-bar 
chart, or pipe-organ chart, than the staircase curve itself, be- 
cause it has retained the shaded areas of the bars. 

The band-chart becomes interesting when it is broken up 
into several bands running together across the page, each band 
representing a component part of the total amount under the 
curve. This is the band-chart proper, a series of layers or 
bands going across the chart which, when taken together, 
form a total whose fluctuations are shown by the curve of 
the top edge of the top band. This chart is sensational and 
interesting but of little precise value. You will find it hard, 
for example, to measure the width of any band except the 
lowermost. In fact the various curves which mark off these 
bands one from another have no value except that the lowest 
curve is a curve of one segment, the second curve is the curve 
for the total of the first two segments, the third is the total 
for the first three segments, and so on up to the top curve 
which is the total for all segments or the whole phenomenon. 


FRENCH WOMEN<WORKERS DURING THE WAR 


Proportion of Women to Total Mmployees in rrance 
1914-1920 
(Source:= Monthly Labor keview) 


an 0 © ° o °o wo o oOo 
MH on 4 o oa ay : So o 
ro 6 o Lo) eo $ 38 83 
ao wn ° nn - wo rt 6S. 
° ° ° e ° wo yee 

MEN oOo wo o & i 
0: 25 6 n wn m2 wn » 


= 


1916-Jan 
1927-Jan 
1918-Jan 
1919-Jan 
1920-Jan 
1920-00t 


Fig. 193. A Relative (or Percentage) Band-chart. 


214 CHARTS AND GRAPHS 


And you will find that area conceptions are inevitably in- 
volved in this chart; the reader tries to measure the value of 
the various segments by the width of their bands. And 
unless staircased these areas will be extremely deceptive, the 
bands appearing to be narrower whenever the neighboring 
bands are moving rapidly up or down. 

The most useful form of band-chart is the “100% band- 
chart.” In this case the entire space between the zero or 
base-line and the 100% line is filled with various bands, each 


CLASS ALIGHMENTS OF THE POPULATION 
Divided into Capital, Labor, and Public 
United States 
1870-1910 
(Source;- Arranged from Census by A. H. Hansen) 


Capital tak 7.7 |10.2 10.8 13.8 

Public 68.2 63.7 48,1 45.4 41.9 

Unolassified 8.1 8.2 9.3 8.5 6.0 

Labor 26.6 30.4 32.4 35.3 38.2 
100 


20 


30 YT. 


10 


0 
1870 1880 160 1900 1910 
Fig. 194. The Smoothed Relative Band-chart. 


COMPOSITE CURVES 215 


THE NATURE OF EXPORT GOODS 
Demestic merchandise exported classed as consumers’ or producers’ goods 
United States 
1910-1919 
(Source:= U. S. Statistical Abstract) 


ieee 5 . . 1 i, a . , ” 
af Re ete wer aes SN 
Po oats Bar REP eee So 8 est 
i o © © - 3 ra © © o o 
PE NCATE CET Pe Hae ah ana en 
F esi M Ss asta Bae Sh Sg gg! oly 
eo 
CRO MASS eS eS TS SLE 2 Ee 


90% fam 
ee fi See 


of Re en a 
a 1 | ae Ze 


wl bes bd AMI 
aay the 


408 CALL LAIN 7) Ye AYY, 
mmceitcall ~~ Ch 


| A LS TTT 


v 
Cal 
a 
Cal 


fo) et nN C2) uw o ~ @ rey 
~ nl ct - — et Cal ct rt 
a cor) fer) bez) fey) Q 2) be7) co7) 
rt oad we Lend a et - to) we 


Fig. 195. The Staircased Relative Band-chart. 


indicating a portion of the total or 100%. The fluctuation 
and changes of these bands show graphically the changes of 


216 CHARTS AND GRAPHS 


the component elements of this 100%. This type of chart is 
often used to show the changes in the distribution of cost and 
profit in an industry. The optional illusion of narrow bands 
when nearby bands are moving rapidly up or down is to some 
extent eliminated when the band-chart is made with stepping 
or staircase outline instead of smoothed polygon outlines, For 


IMPORTS INTO THE UBITED STATES. 
Value 


1920 ; 
Percentage from the different continents 


KO OOO 
RKC RRR 

nas 08 ERR SRD 
reset SKE Noten 


Africa o «a Cr, a i oe oe od) 
Oceania ©? « a aaanroe«#dnereanewseaevween 8 @ 
Asta 3 eo a Aa nM I: Ta ae Nels gh IS TM Sk 
Pseieeas oie POMACR BEE SANS ae aa 
erica 

fsb ties <4 # SS ARRSSRSRARARARRESE 
rica 

Burope @ 8 2*8 8 S288 8RB5S 8RB5E 

00 - = “5 = 


rr 


\ 


ane 


N 
AANA 


aa 


ZA 


Uy 
WAY tii 
rae Wa 


\I 
WS 


N 


NN 


= 


ANS" 
WS IGG FY 


\ 
W 
W 
WY 


LAA 


a simple presentation of the changes in the component parts 
of any phenomenon, this 100 per cent stepping band-chart is 
admirably suited. It is extremely popular in its appeal and 
does not suffer from the general disadvantage of staircase- 
curves because there is no question of superimposing other 
similar charts. It is the right way to represent cost compo- 
nents and other percentages to a general public, being within 
its narrow limits, safe, sound, and attractive. And the reader 
will notice that it is a form of curve which is well-nigh indis- 


COMPOSITE CURVES Sui 


BXPORTS FROM THs UNITED STATES 


1800 = 1920 
Percentage to the different continents 


eo ° ° a a tt Ht HF eH Ht HR HM MM ltl 
Oceania @ ° ° ° ° on AN HM HR A a au 8 mM Wm Mm 
Asia % « © a oS) es @¢ e@ wae Ne t HNN Dw oO Se + @ 
South oO x n ) r") ou vw ot fF ewe fF wD fF Mm fF wD e © 
Amsrica 
Bf ] 2° 

” 3% BY 2 a Ry testy Ay EE} OM Ey eel ECG eG 4 
Amsrica 
Ponce crepes ee aee ©) UP Relea Ss Sos 68 e S46 ses 

FRIOA 98 <0: a 

x} = a"! % wu 

SO 


y 


ZA 
Zi 


N 


SSESSS0 


NN 


ZY y 
aoeee! Vy, oe 


Dilla 


N 


Li) 


Fig. 197. 


YYYY 


tinguishable from a bar-chart, being virtually a vertical-bar 
or pipe-organ compound, relative bar-chart, or in other words, 
a series of 100% bars set on end and brought into contact with 
each other. 

In addition to the Rorecoine more or less distinct types of 
curves, there are also many and various possible embellish- 
ments which belong to the field of artistic rather than that of 
statistical endeavor. The object which is being charted may 
be pictured realistically and the picture shown at the end of 
the curve. Indeed, the same picture may be used frequently 
along the curve, or the picture may be modified to reflect the 
changes which are shown mathematically by the curve. 
Several different pictures may adorn as many different curves, 
and where one rises particularly high, it may be given a pair of 
wings or set in a balloon or aeroplane. By these and other 
fanciful ways, the imaginative chartmaker may make the 
appeal of his chart more vivid. But such measures are out- 


as 


, (i i 


Y 


S 


- = 


= = a 


NNN span 


Fa) SSS l ST i 

2 SN AEE SS FM ES SSN TSH aE pe 

Fiz. 138. The Relative Chart is is Supplementary. 
Te is well to © Show data of this Lind by two charts, the absolute and the relative 
Per ast Gsumbution, and the Latter can well be smaller—From Joseph E. 


gue, Eomomas of Pro 


COMPOSITE CURVES 21g 


150 


1915 1916 1917 1918 1919 1920 1921 


U 


CHANGES IN THE STANDARD OF LIVING. 


Index Numbers of Weekly Earnings in New York Factories, of the Cost of Living in 
the United States, and of the Living Standard (‘‘Real Earnings’’). 


Fig. 200. A Pictorial Curve. 


side the proper scope of this book; the pictorial curve, like the 
pictorial bar-chart, is really intended for, and is useful for, 
popular consumption. We have come so far into the subject 
of mathematical charts that we shall hereafter have no time 
for purely pictorial effects. These may be left to the enter- 
prise of the individual. 


CHAPTER XX 
HISTORICAL CURVES 


Statisticians divide all series of figures into two groups. 
A series involving time, that is, a series for which different 
points or periods of time are the stubs or independent variable, 
they call a historical series. Other series in which time is not 
the independent variable, they call frequency series. This is 
a convenient classification for the chart-maker, and we can 
therefore divide all curves into historical curves and frequency 
curves. The historical curve has by some writers been called 
the “histogram,” or “‘historigram,” and the frequency curve 


NEW INCORPORATIONS 
Capital Invested in New Enterprises 
Whose authorized capital equaled or exceeded $100,000 
Principal States, U. S. 
1919-1921 
(Source:- N. Y. Journal of Commerce) 
(Figures in millions of dollars) 


1919 1920 1922 
Jan 492 2280 1243 
Feb 324 1159 654 
Mar 371 1376 955 
Apr 616 1354 988 
May 749 1418 601 
Jun 1255 1323 676 
Jul 1420 1260 282 
Aug 823 941 580 
Sep 1947 951 490 
Oct 2364 1160 503. 
Nov 1341 896 368 
Dec 1078 861 619 

Fig. 201. A Historical Series. 
220 


et 
nb 


HISTORICAL CURVES 201 
the “pictogram,” but these names have not been widely ac- 
cepted in a precise sense. 

In historical curves time is always the x-variable and must 
be plotted on the horizontal axis, its divisions forming the 
x-scale. In a previous chapter on plotting points the need 
for a precise scale has been discussed and the two methods of 
indicating periods of time, either by points on the scale or by 
spaces between points, have been dwelt upon. The reader is 
urged to review this section, as it meets a serious problem in 
the plotting of complicated historical data. The reader is also 
referred to a previous chapter on curve scales, in which the 
most useful forms and positions of the chart-field were ex- 
plained. 

The field of a historical curve-chart should be positioned 
very close to the righthand edge of the sheet of paper on 


Fig. 202. Year by Months, Universal Ruling. 


“Aiseq OuIquod szzeYyD [enprarpuy eyL “£0 “314 


o = 
o °° 
eo o< 


3 
oe 


n 
° 
Ks) 


> 
£ 
A 


Ts6 


o 
° 
o 
- 


eze'Tt 


ree» 2 
ee wb Db 
¥r¥ OaNQ A © 
os © 2 oO 

00"00T$ P 


8L0‘T 


616 


00° 00T? o} 


STé: 


8 

ry 

_a 
GseTTOD Jo GUOTTTIR 


(SIETTOP JO SUOTTITs ay seundtyz) 
(aot0umog Jo Teusmor *k *N -:90EN0S) 
, Lest atet 
"S *h 6899815 Tedroutsg 
@50°00TS Pepesdsxe Jo poTenbs teytdso baztioyy 


ne 280m 


Sespidiequg aoN UT payseaul Teyydep 


SNOILVYOdUYOONT BIN 


HISTORICAL CURVES 233 


which it is drawn. There should be not more than a quarter 
or half inch margin between the chart-field and the edge of the 
paper on this side. The reason for this position will be clear 
to you the first time you prepare a series of historical curves 
in which the curve travels across sheet after sheet of paper 
through a succession of years or periods of time. By over- 
lapping the charts (fanning them out) so that they are all vis- 
ible, with only these narrow margins between them, the entire 
series can be made to appear as a single chart. In this form 
the entire series can be conveniently studied and econom- 
ically photostatted. The narrow margin between each chart 
serves to break up the curve into its component periods with- 
out destroying its continuity. In this way a chart many feet 
long can be made upon ordinary sheets of paper without past- 
ing them together and without inconvenience in filing or 
handling them. And from this long series, a single chart for 
a single period can be abstracted and individually compared 
with other individual charts. The narrow margin to the right 
of the chart may at first seem surprisingly inartistic, but it 
pays for itself in the flexibility of uses which it gives to the 
chart. 

For a similar reason it is well to place the field of a his- 
torical curve-chart as low upon the page as possible. You 
must of course have room to enter the figures for the horizontal 
or x-scale legibly. This will rarely require more than three- 
quarters of an inch. By placing the field low upon the page, 
it is possible to compare curves for similar periods of time by 
laying one directly above the other, overlapping them ver- 
tically so that the two curves are both visible and their ordi- 
nates and time scales coincide. By this device, the reader can 
easily compare the seasonal or periodic fluctuations of two 
curves and at a glance detect the extent of their similarity. 

In short, the field for a historical curve should be as close 
as possible to the lower righthand corner of the sheet of paper 
upon which it is drawn. This leaves a very large margin at 
the top of the page above the chart, in which should be en- 
tered the data of the curve. It also leaves a large margin to 
the left of the chart. This margin will be partly filled by the 
important vertical or y-axis scale or scales (if two or more 
scales occur on the same chart). But the chief use for the 
lefthand margin is that in it can be written the notes, comments 
or explanations which may be desired with the chart. If the 


24 
NEW INCORPORATIONS 

Capital Invested in New Enterpr‘ ses 

Mose authorized capital equaled or exceeded $100,000 
Principal States, U. S. 
1919-1921 
(Source:- N. Y. Journal of Commerce) 
¢Pigures in-millions of dollars} 


womy 82996833338 


@illions of Dollars 


£92, 


Uilitons of Dollars 
= 
NI 
Gee £ 
meas. 
Revs 
Fae 
aa 
Da ose 
ee ee 
a 


1920 


2000 


Millions of Dollere 
rd 
a 
$ 


» 
° 
oS 
co 


ml 


1919 


tov 


Fig. 204, Fanning Up and Down to Compare Seasonals. 


HISTORICAL CURVES 225 


sheets are to be bound in a loose-leaf holder or book, the 
binding edge will be on the extreme lefthand edge still further 
away from the chart. At the top of the page above chart and 
data, the title should be placed. 

In historical curves, possibly more than in most, it is im- 
portant that the data appear with the chart. It is important 
for the maker of the chart, for the curve is more easily plotted 
direct from the data, and the plotting checked for accuracy. 
It is important for the reader who is thereby enabled to either 
satisfy himself as to accuracy or to find any particular value 
without relying upon approximations more laboriously de- 
ciphered from the scale. The proper position for the data is, 
as has been said, above the chart, each value plotted appearing 
on line with the ordinate of its plotted point. Unless the data 
is extremely simple and brief and can, without crowding, be 
written horizontally, it is better to enter it vertically, writing 
or typewriting on edge, in the manner described in a previous 
chapter. 

In historical curves, we have much use for a few simple 
mathematical and accounting phrases. The first of these is 
the “cumulative,” or “total to date.” When beside a column 


NEW INCORPORATIONS 
i Capital Invested in New Enterprises 
Whose authorized capital equaled or exceeded $100,000 
Principal States, U. S. 
1919-1921 
(Source:- N. Y. Journal of Commerce ) 
(Figures in millions of dollars) 


|Monthly | Cumula-|Monthly | Cumla-} 
| tive | tive 


2,280 | 
3,439 | 


4,815 | 
6,169 | 


7,587 | 
8,910 
10,170 
11,111 
12,062 
13,242 
14,138 
14,999 | 


Fig. 205. Sirnple Series and Annual Cumulations. 


226 CHARTS AND GRAPHS 


of figures showing the sales of your company, month by month, 
you place a second column of figures in which are entered the 


NEW INCORPORATIONS 
Capital Invested in New Enterprises 
Whose authorized capital equaled or exceeded $100,000 
BB aaa arta U. S. 
91 


(Source:=- N. Y. Journal of Commerce) 
(Figures in millions of dollars) 


uN wm oO er wo + O© Ff 
cmistive = $888 88 B38 BB 
a) 8 a Ca oe. “@: 7 @ © @ 2, @, )8 
xa AFA NAN AaeeasUOUka_ Ot ep 
NN At on nYM MW FP FF A 
Monthly a88se3ese28ees5 8 
10000 


Millions of Dollars 
ns 
8 


Fig. 206. Series and Cumulation Plotted With Same Scale. 


——e' .! 


HISTORICAL CURVES 207 


total sales to date this year, your second column of figures is 
a “cumulative.” The cumulative series begins with zero at 
the beginning of each period of time, that is, before the first 
value in the period, and is built up by adding each item to the 
previous cumulative until the final entry of the cumulative 
series is the total for the entire period.! The cumulative for 
the next period then begins with a new zero and similarly 
builds up to the total for the next period. And it is important 
to remember that at the end of each period, the cumulative 
has mounted to and equaled the total for the entire period. 
We deal here, of course, with periods of time (such as years) 
which contain a series of individual values for shorter periods, 
such as months. 

Not all data can be cumulated. Economists make a dis- 
tinction between “stocks” or “funds” and “streams” or 
“flows” of goods or money. A stock or fund of goods is some- 
thing which can be considered in existence at a certain moment 


TI ks 40 
42.90 
43.40 
43.68 
44.08 
44,80 
47.16 


gaeregze?e 


49.12 
Bep 60,46 
Oct 49.23 
for 41.28 8 
Deo 36.06 
£622 Jan 33,96 
Tod 33.46 
Mar = 28,16 
apr = 26.98 
May 26,28 
jun 24,79 
Nl 22.84 
aug 23.98 
Sep 21.98 
Oct 21.96 
Kor 21,90 
Yeo 21.98 
JOLESALB PRICES OP 
|BESSEYER P10 IKON 


(per long ton) 
(Qource:- Bureev of Labor Starvericn) 


Fig. 207. This Data Cannot be Cumulated. 


ee ae ee eee 

1 By the forward cumulative described in the text we obtain “up to and including” 
figures. It is of course possible to cumulate historical figures backwards, obtaining 
“after and including” figures, but the step seems purposeless.—Cf. Secrist, Horace, 
An Introduction to Statistical Methods, pp. 232, 267, 


228 CHARTS AND GRAPHS 


or instant of time, while a stream or flow of goods is something 
which takes place during a given period of time. Figures of 
the latter, that is, stream or flow figures, can be cumulated. 
Obviously, if sales have continued throughout the year the 
sales for each month can be cumulated, that is, can be added 
together to give a total of sales for any period of several 
months or for the entire period of the year. On the other 
hand, the figures of a stock of fund or goods cannot be cumu- 
lated. In business, a common example of a stock or fund is 
the stock on hand or balance at any point of time. And 
obviously, if your balance was $3,000 on the first of January 
and $5,000 on the first of February, you cannot speak of your 
balance for the two months together as $8,000. It is not 
difficult to decide whether a series can be usefully cumulated. 
The use of cumulations or series of sub-totals is frequent in 
accounting. 

The next mathematical conception is at present little used 
in ordinary accounting but is far more valuable for most ana- 
lytical purposes than the cumulative. It is called the “‘moving 
total.” To take the example given above, if beside your 
figures for the monthly sales of your company, you were to 
enter another column of figures showing the sales “for the last 


NEW INCORPORATIONS 
Capital Invested in New Enterprises 
Whose authorized capital equaled or exceeded $100,000 
Principal States, U. S. 
1919-1921 
(Source:=- N. Y. Journal of Commerce) 
(Figures in millions of dollars) 


1919 1920 1921 


I— 

Monthly] Monthly | Moving | Monthly | Moving 
Annual Annual 

! Total Total 

= 


2,280 | 14,468] 1,243 | 13,962 


1,159 | 15,303 654 | 13,457 
1,376 | 16,308 955 | 13,036 
1,354 | 17,146 988 | 12,670 
1,418 | 17,815 601 | 11,853 
1,323 | 17,883 676 | 11,206 
1,260 | 17,723 282 | 10,228 

941 | 17,841 580 | 9,867 

951 | 16,845 490 | 9,406 


1,180 | 15,661 503 | 8,729 


896 | 15,216 368 8,201 
861 | 14,999 619 7,959 


Fig. 208. The Simple Series and Its Moving Annual Total. 


HISTORICAL CURVES 229 


twelve months,” this new column would show the moving 
totals. Beside the January sales in 1921, you would enter the 
sales of the twelve months beginning with February, 1920, and 
ending with January, 1921. Beside the February, 1921, sales 
you would enter the total of sales for the twelve months 
beginning March Ist, 1920, and ending February 28th, 1921. 
It is easily seen that a moving total can be carried all through 
the year by simply taking the total sales for the previous year 
and successively subtracting the sales for the thirteenth month 
back, and adding the sales for the last month. Each figure in 
the moving-total series can be obtained by dropping off one 
month in the earlier year, and adding the corresponding month 
in the later year.? 

The moving total is so useful that it has sometimes been 
enthusiastically described as the balance-wheel of commerce. 
When you are plotting the curves month by month, you are 
apt to find a considerable amount of monthly fluctuations, 
due in part only to normal seasonal conditions. In such 
cases, these perfectly normal seasonal fluctuations may hide 


2 The work-sheets for computing moving annual totals should always be designed 
to show similar months (or other parts of the cyclic period) together. This is done 
by arranging the periods in successive lines (with like months below each other) or in 
columns (with like months beside each other). Space should also be left for two other 
figures in each month: the first of these is the moving annual change, that is, the 
algebraic difference between the two like periods; the second is the moving total, 
that is, the cumulative of the moving annual change. . 

A second method is to arrange the monthly “(or original) data in one long column. 
The comparison between like months may then be effected easily by means of a 
movable slip of paper with two slots or windows at the appropriate places to make 
the two desired months visible and hide all intervening months. The moving change 
and the moving total figures can then appear (each in a column) in two columns 
beside the column of original data. Totals should be taken directly from the original 
data at intervals for checking purposes. 

If the cumulative also is being computed, still a different form of work-sheet is 
useful. In parallel columns the successive years should be tabulated, with each 
monthly figure on every fourth line down the page. The sheet should then be fed 
into a listing machine and the tabulated figures reprinted immediately below their 
entries. As they are listed, sub-totals should be taken on every record line below 
them, these forming the cumulative series. The sheet is then removed from the 
machine and the moving annual total entered by hand from a calculating machine 
(which adds and subtracts the proper months from the last totals and retains the 
results), these being entered in the remaining blank line for each month. This paper 
should be originally ruled with horizontal faints at listing machine intervals (of one- 
sixth of an inch) and with horizontal heavy lines every fourth line to separate the 
months, the page having 48 lines in all. As a further guide, vertical faints can be 
ruled in at listing machine intervals (one-sixth of an inch). If not specially printed 
up to order, the cheap cross-ruled paper with lines every sixth of an inch can be used. 
These details all tend to make checking up for errors very simple, and largely eliminate 


mistakes. 


230 CHARTS AND GRAPHS 


or obscure the true trend of the business, or at least make the 
determination of the trend more difficult. But you can easily 
tell whether the general trend of your business is upward or 
downward by plotting the curve for the moving annual total. 
The moving total series appears to flatten out the seasonal 
fluctuations and respond only to the true movements of the 
trend. In fact the moving total is sometimes called, even by 
statisticians, the “‘trend.”’ 

| The moving total need not be annual, but can be computed 
for any given period of time. Thus we may have a moving 
24-months total, or a moving 5-year total. In any case we 
have again periods within periods, as in the cumulative. The 
most usual form is the moving annual or 12-months total, 
for ordinarily in business there is a certain amount of normal 
monthly or seasonal fluctuation which repeats itself every 
year. These annual seasonal fluctuations are naturally swal- 
lowed up in a total for twelve months, for such a total always 
includes every month in the year. The moving total is in 
general an excellent device for smoothing out the wrinkles 
and wiggles in a curve and reducing the curve to a simple 
regular trend-line. It should be used whenever the cycles of 
fluctuations appear to be of regular and uniform length or 
periodicity. 

A word of caution is necessary about the plotting of a 
moving total. Strictly speaking, each item in the curve should 
be plotted in the centre of the period which it covers. Thus, 
the plotting point of each figure in a moving annual total series 
would normally be midway between the ordinates of the sixth 
and seventh months covered by the figure. The entire period 
of the total being one year, each point should be placed in the 
middle of the year which it represents. In this case, the moving 


* Whenever the period of the annual cycle is not regular, as in crops and tempera- 
ture cycles (one period of 124% or 13 months, the next of 1114 or 11 months) it is 
well to follow Professor Secrist’s suggestion of a thirteen-month moving total. This 
has the further advantage of centering the moving total figure precisely upon a monthly 
one (the seventh) instead of midway between two monthly ones (the sixth and seventh). 

The same advantages are much better secured by an average of an eleven-month 
and a thirteen-month moving total, both centered on the same months. This may 
be called a “taper-smoothed” eleven-thirteen-month moving total, as it gives full 
weighting to the central eleven months and half-weighting to the terminal months 
(first and thirteenth). It will be seen that this precisely corresponds (in the average) 
with the periodicity of eleven to thirteen months. The taper-smoothed eleven- 
thirteen-months moving total is easily computed from the twelve-months moving 
total, as it is the two-months moving average thereof. Of course, a longer taper can 
be used if desired. The test is smoothness of the resulting curve. 


HISTORICAL CURVES 231 


total curve will begin five and one half months after the 
beginning of the curve of individual months, and will end five 


NEW INCORPORATIONS 
Capital Invested in New Enterprises 
Moose authorised captial equaled or exceeded $100,000 
Principal States, 0. S. 


«Journal of Commerce) 
(Pigures in millions of dollars) 


e 

tor meet mee SPSS 2P RRS ERASER EASE Raa 
3 7 eS eee ee ee OS SON ONE ORO OHS LCR EE AS OTS OS. ete 
2 
2 
ja Tetgurrent 22S P LIRR ESERIES ELAZES ERAS 
Oa ad SSS ses ie da dciddadsceesag 
a J 

tergoning § 2 9 8 8 $€2£23 98 & GER RES 

nese See erent nnaddwaadidann se woe 


Monthly 


Millions of Dollars 


PRE 
y, 


Fig. 209. Three Positions for the Same Moving Total. 


and one half months before the end of the monthly curve. 
The moving annual total may then be called a “total for the 
current twelve months.” For certain purposes, however, you 
may desire to have the moving total end on the same ordinate 
as the monthly curve. This can be done by plotting each item 
at the end of the year which it represents. It has the advan- 
tage of giving a more up-to-date appearance to the chart, but 
it is now necessary to label the series moving “total for past 
twelve months.”’ On rare occasions, you may desire to place 
the moving total at the beginning of its period, in which case 
it becomes a “total for the following twelve months.” When 
comparing trends, however, between various items, it is 1m- 
portant that these moving totals be plotted at similar points 


CHARTS AND GRAPHS 


NEW INCORPORATIONS 
Capital Invested in New Enterprises 


Whose authorized capital equaled or exceeded $100,000 


Prinoipal States, U. S. 
1918 


(Source:= N. Y. Journal of Commerce) 
(Figures in millions of dollars) 


eaasSanak®tiKR Pls S 

Moving Total for wm = ao OD 3S LOS OF Oke se 
Previous Yoar wt Ma gn st ow Ot US OM 
$ERSEezRsSRagsE 
Moving Total for Pers Ot en ek ee i ees 

Current Year S Se Pra Wits) Rad ake. ak ee ete ne 
mo hm om A 2 A a a ew a Oo wo 
ing T 6228 8 ® 83S3& 8 
Wolledligtees Here. Syne eee 
es 

ng a nr Nn © i) ~~ Ww o© @ Oo rt 

a nc 

en > o @ fF 08 HOH H YY a 

Mont Fa ano ok Ot FF Hm 
minty Sea Re area Ore aia 
- 


2,399 


Fig. 210. A Detail of the Last Figure. 


ie 


HISTORICAL CURVES 233 


in their periods, else an unwarranted lag will appear between 
the fluctuations of the two charts.4 

Similar to the moving total is the “moving average.”’ The 
moving average is merely an average secured by dividing the 


Moving | Moving Moving | Moving 
Total | Average || Total | Average 
| 14,468 
15,303 
16,308 
17,146 


| 17,815 
| 17,883 
1 17,723 


17,841 
16,845 
15,661 
15,216 
14,999 


NEW INCORPORATIONS 
Capital Invested in New Enterprises 
Whose authorized capital equaled or exceeded $100,000 
Principal States, U. S 
1920-1921 
(Source:+ N. Y. Journal of Commerce) 
(Figures in millions of dollars) 


Fig. 211. Meving Annual Total and Average Series. 


moving total by the number of items which compose it, that 
is, a moving monthly average is secured by dividing a moving 
annual total by twelve.5 The moving average has the great 
advantage of lying at about the same height on the chart as 
the curve of individual periods (e.g. months), from which it is 


4Needless to say, the only accurate picture of events (as regards time) displayed 
by the moving-total or average curve is that shown by the curve of the current 
twelve months or other period, that is, the curve of points plotted at the centers of 
their periods. (Cf. Chapter on Plotting Points, supra.) This is the true smoothed 
curve. At all other positions, the curve has been arbitrarily “lagged” forward or 
backward. 

5 The work-sheet for the moving average is the same as that for the moving total 
already described, save that an additional space must be left in each month (in the 
second, or columnar, method, it would be an additional column) for the moving 
average, which is derived from the moving total by dividing the latter by twelve 
(annually, or by whatever the number of items be which go to make up the total). 
(Obviously in the taper-smoothed eleven-thirteen-month moving total, the two 
terminal months have only half weight and the total is still of twelve months.) 


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derived.6 And for this reason, you will find it an even better 
method of smoothing curves and showing their true trends 
than the moving total.’ 


a 
6 The moving average has another great advantage over the moving total, in that 
it can be given variable period-lengths to conform to cycles of variable lengths. 
This is not possible with the moving totals. 
"The moving totals and averages are also sometimes called “progressive” totals 
and averages. 


CHAPTER XXI 
CYCEES 


Long ago, or was it yesterday, there were neither auto- 
mobiles nor aeroplanes, and the streets were frequented by a 
cheery and wholesome class of persons, who conveyed them- 
selves about on two-wheeled contrivances called bicycles. In 
deference to our age, the reader will permit us to pause and sigh 
a moment over this happy retrospect. Sometimes in the circus, 
a contemporaneous antiquity, trick riders rode one-wheeled 
affairs with perilous skill. Needless to say, the rims of these 
wheels were as smooth and regular as the circumference of the 
average clock-chart. | 


pate. O-L 7-22 ) poner no // 


2 fe) 
finewan A-fernand) oe 


FIREMAN. .. . 


Permission of Mr. Walter 'N. Polakov 
Fig. 213. The Mechanical Cyclograph. 


235 


ey ee 
*; 
J ~ 


Now a clock-chart—if you have forgotten your early 
chapters in this book—is round like the face of a watch. 
Radiating lines or radii take the place of ordinates, and con- 
centric circles or rings take the place of abscissae. Hence the 
chart can be used for the display of recurrent data—that is, 
historical data which after a certain period of time repeats 
itself. Your only care must be that the period of the cycle 
be adjusted evenly and wholly in one complete revolution 
about the circle. Then the curve of the data will meet at the 


936 CHARTS AND GRAPHS 


Seasonat Fructuarion on Buitoine Operations 
AVERAGE YEAR, 1910-1920, on 25 STATES 
(Source:e= F, W. Dodge Co.) 


Fig. 214. Asa Chart, This is Worthless. 


— 


CYCLES 237 


two ends of the period, forming a continuous and endless 
curve.! 

Had the trick cyclist in the circus used a wheel the rim of 
which followed the uneven outlines of this curve, he would 
indeed have had a bumpy ride. And a line drawn on the wall 
behind him, following the shadow of his head, would mark 
the same curve plotted on a chart-field of plain co-ordinates. 
Study the curve as so plotted in the ordinary way, and you 
will see that once every so often the wiggles or fluctuations 


SEASONAL FLUCTUATION IN BUILDING OPERATIONS 


Total for 25 States in average year 1910-1920 
(Source:- F. W. Dodge Co.) 


apie Se Beck: Re iets 
Se ee Ge aie tee coe Weer et eae 
Dollars: g ¢ a g 5 a $ $ a 8 3 8 
, =the dS) 
350,000,000 
300,000,000 | | 
.250,000,000 
200,000,000 
150,000,000 
100,000,000 
50,000,000 
0 


Fig. 215. The Rectilinear Co-ordinates Are Much Better. 


a eee ee 
1 The clock-chart is similar to carp, the fish which is properly prepared by throwing 
it away after it has been cooked. When the clock-chart has been well and carefully 
drawn, it is ready for the waste-basket. For this reason, no detailed discussion of it 
or its polar co-ordinate field is entered into. The only case in which the clock-chart 
is a justifiable product is the case of automatic mechanical charts or cyclographs. 
These are parts of recording machines for temperature, pressure and the like, in 
which a fountain-pen at the end of a pointer leaves an inked record or curve upon the 
rotating disc underneath it. They are graphic records, but not otherwise useful charts. 


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SP] te oO 1G SOs 
a OT FO HU 


1913-1921 
(Average 1913 © 100) 
(Source: Bureau of Labor Statietics) 


United States 


PRICES OF EOos 
Relative Figures of Retail Prices of Eggo 


1922, 


19290 


isis 


1s1e 


1917 


1916 


2915 


isie 


1915 


239 


Cycles of Slightly Varying Lengths. 


Fig. 217, 


240 CHARTS AND GRAPHS 
cyclic fluctuations, and to be able at will to remove them. 
The subject has already been touched upon in the last chapter. 

The time between the commencement of one cycle and that 
of the next, is called the period of the cycle. This period may 
be short or long, according to the nature of the data, a very 
frequent short cycle being that of 24 hours or one day. In 
business statistics, there are cycles longer than a year. Some 
investigators have found evidence of business cycles in which 
eras of general prosperity, depression and crises repeated 
themselves every four, eight, or even twenty years.2 _In meteor- 
ological and astronomical sciences, cycles of dry and wet 
weather have been found to last thirty-three years and of 
warm and cold weather about a hundred years. The most 
important cycle in most business statistics is the annual one, 
of four seasons or twelve months. The student of business 
statistics can almost always assume that he will find more or 
less of an annual cycle of seasonal fluctuations. Sales may 
repeatedly rise in spring and fall and decline in summer and 
winter. Production may then fluctuate somewhat earlier in 
the year, anticipating the changing demand. If production 
is uniform, ware-housing cycles will appear in order to absorb 
the surplusage in low selling periods. 

It is not often that the recurrent cycles are identical either 
in the shape or the height of their curves. Such variations 
may be due of course to incidental and insignificant causes, 
but in general studies of broad trade or economic movements, 
they are often given a more fundamental importance, as being 
significant of real changes in the phenomena studied. The 
problem then is to isolate these variations in the cyclic fluctu- 
ations, that is,-to eliminate from the series its seasonal cycles 
and retain its significant changes. To the series which remains 
after the removal of cyclic fluctuations, the name of “secular 
fluctuations” if often given. And we may therefore look upon 
the original series as being a combination of two different sets 
of forces, or movements, which we call, respectively, cyclic 
change and secular change. Either or both of these elements 
may be the object of your study and it is important that you 
should be able to determine them easily. 


2 The literature on this subject is considerable. In particular, the student should 
refer to: 
Mitchell, Wesley C., The Business Cycle, and 
Moore, H. L., Economic Cycles: Their Law and Cause. 


- Be ae Roa 


GUCLES” ~ | es 


SEASONAL VIRULENCE OF SCARLET FEVER 
Number of Cases reported to Boston Board of Health 
1900-1904 


Fig. 218. Showing the Use of Relative (Percentage) Figures 
and a Rounded Curve. 


Indeed, the statistician, and the chart-maker as well, would 
fail in one of his most obvious tasks, if he were to report as a 
significant rise or fall, a change which was wholly due to its 
cycles. Are we to conclude that the telephone business is dis- 
appearing because in a series showing hourly number of phone- 
calls, our last report is the number of phone-calls between 
twelve and one o’clock at midnight? It is true that between 
mid-afternoon and mid-night the telephone activity has 
dropped off almost entirely, but we must remember that it 
does this every night (with the possible exception of election- 
day) and that we deal here with a daily cycle. Are we to 
conclude that the cold-storage of eggs is a practise of the past, 
because our monthly report of warehouse stocks end with 


242 CHARTS AND GRAPHS 


ACCIDENTS 1N MANUFACTOR1WO 
Hourly Occurrence of 364 Patel and 21,461 Won-fatal Aooidents 
Illinois 
Three Years, 1010-1912 
(Percentage Figures Only) 
(Source;- United States Bureau of Labor Statistics) 


2 
. - 
2 


Ree © 
byes 3 7 2 


posthe 


Percentage of accidente 
S 


Night Oay Wight 


Fig. 219. Daily Cycles. 


March, when as a matter of every-day knowledge there is 
an annual cycle and the stocks are always low at this time of 
the year? 

The subject is more properly a statistical one than a chart- 
ing one, but it is of such importance in the making of the © 
specialized form of charts which follow that we will outline 
briefly some of the simpler methods in use. Our concern here 
is with the separation or elimination of the recurrent or cyclic 
fluctuations in an historical series. Ordinarily, in business, 
the seasonal fluctuation, that is, the annual cycle, is most im- 
portant, and the following explanation will be limited to it. 
Other cycles may be similarly treated. The most elementary 
consideration in the analysis has been made obvious by the 


CYCLES | 243 


COLD STORAGE HOLDINGS OF BSCS 
Stocks of "Case Eggs” in Warehouses 
United States 
1916-1921 
(Source:- Survey of Current Business) 
(Monthly Average for Five Years, 1916-1920, = 100) 


S yr. Average 
1916-1920 


Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 
Fig. 220. 


foregoing illustrations; namely that the relation between the 
last item in a series and the corresponding item in the previous 
cycle in the same series is of more importance than the rela- 
tion between the last item and the immediately preceding 
item. Inthe last illustration, how do this year’s October stocks 
compare with stocks of October last year, not how do this 
year’s October stocks compare with this year’s September 
stocks. 

But every business man has progressed beyond this ele- 
mentary stage. He asks to see the figures for the previous 
month in each of the last two cycles. For he knows that it is 
more important to see how the change in stocks from Septem- 


hen a 


wa hi uel Tieden Gee 1 


244 CHARTS AND GRAPHS 


ber to October this year compares with the change between the 
same months last year. We may generalize this by saying that 


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we are now concerned with the relation between the change in 
the last two items of the series and the corresponding change in 


CYCLES 946 


the corresponding two items in the previous cycle in the same 
series. Now it is precisely this relation which the moving 
annual total, or average, described in the previous chapter, 
tells us. A little study will show that the moving total swal- 


EGG PRODUCTION 
Receipts of Eggs at Five Markets 
(Boston, New York, Philadelphia, Chicego, and San Francisco) 
United States 
1920-1921 
(Number of Cases) 
(Source:- Survey of Current Business) 


° ° [2} ° ° °o ° ° 
Monthly (3) ke feP ewe ta | Ko) 8 Bias te 1 Ss 
Movin, eas eS eo Sn OS Med et Oe 
‘3 i Q wo or) a oO = wo ™ Nu ” s a 
12-No. a ww OO eo I am TO! et) et ie 
Average Ds Nie SEO Shee ASR he SRG eC 
5 Pe Pee Met et et rd (et ee et) le 
° GEVOEL ORO! KON LO. OO.) OO. Oy ON LO 
Ss SOU SP Ou OMCHEOM OMS! US. (Own Ss OMIOl (Om ONO NOM EO UL OES) 
Monthly ee ON aa ONO CN ROE OP Rc g Oey LO, OOM COlne Rs) Ole Get gO UE DIP Os Stet) ee oS 
Total See oS ena ES La oa te 9 oe ae CO etre gS ey oie Lea Se eas he | 
Sh ee ees Pee oes ie 
a eal sige 
2,500,000 
2,000,000 
1,500,000 
1,000,000 
$00,000 5 


7 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 
1920 1921 


Fig. 222. The Moving Average Shows Trend. 


lows up the cyclic variations by the simple process of swal- 
lowing up all the various items which make up the cycle. The 


3 As a matter of fact, the changes of the, moving total from month to month are 
merely the differences between figures for the new month included and the old month 
excluded (one year previous). Hence, the work of computing a series of moving 
totals can progress largely at sight by comparing the month to be added with the 
month to be subtracted (one year earlier) and algebraicly adding this difference to 


the last moving total. 


246 CHARTS AND GRAPHS 


period covered by the moving total (or average) must of 
course be of the same length as the period of the cycle.4 And 
the resulting changes in the moving total are merely the dif- 
ference between changes in corresponding pairs of months in 
the two cycles. For this reason the moving annual total or 
average may be called the simplest method of eliminating 
seasonal (or cyclic) fluctuations, and determining the true 
secular (or long-time) movement. 

While for many purposes, the trend as indicated by the 
moving total is a sufficient index of the nature of the more 
fundamental changes in the phenomenon, yet in the broad 
study of economic or trade movements, it still retains too 
many insignificant changes. It is true that the moving total 
smooths out all the periodically recurrent fluctuations. But 
it does not yet yield a simple series of perfectly regular change, 
that is, a straight line, or simple mathematical curve, which can 
be expressed by a mathematical equation or summarized in so 
simple a statement, as that, for example, “the population gains 
two per cent annually.” In a much more precise sense the 
latter, that is a fitted straight line, parabolic curve, or other 
regular series, is called the “secular trend.” It is also some- 
times called the “normal” for the particular curve. 

Fitting a straight line, or regular curve, to a historical 
series, is-a matter of mathematical statistics into which we 
need not go, for it requires skill and judgment to which no 
simple rules of procedure apply.® It is sufficient to say that 
when this is taken into consideration the original series of data 
which we are analysing can be considered a combination of 
three elements, namely seasonal or cyclic fluctuations, a secular 
or normal trend, and secular fluctuations. And in the best 
statistical work both the former are often removed from the 
data before the curves are published. When this is done, the 
reader is advised of it by a simple statement to the effect that 
the figures published “‘are corrected for seasonal changes and 
normal growth.” Fortunately he has no idea of the problems 
involved in this correction. 

Accepting then, the moving total or average, as a satisfactory 
method of smoothing away all the insignificant and periodically 


‘When cycles are of varying lengths, this does not apply, for the moving total 
can only be made with uniform lengths or spans. The device next mentioned, how- 
ever, the moving average, does not have this limitation and can be made co-extensive 
with the cycle. 

® Cf. Chapter on Curve-Fitting, 


eS  eeE—eEeEeeEeEeEeEeeeeeee 


CYCLES 247 


recurrent fluctuations which often make monthly curves un- 
satisfactory—a means in short by which we can promptly plot 
the trend or general direction of underlying movements in an 
historical series—we turn to the question of determining the 
true nature of the cyclic, that is, the ascertaining of the true 
seasonal fluctuations. We wish now not to eliminate the cyclic 
changes in the data, but to eliminate everything else in the 
data and retain the cycle alone. How can we isolate the cycle? 
The simplest method and one whichimmediately suggests itself 
is to take a single cycle and forget the other cycles in the data. 
This gives us beyond peradventure the change within the 


NEW INCORPORATIONS 
Capital Invested in New Enterprises 
Whose authorized capital equaled or exceeded $100,000 
Principal States, U. S, 


1 
(Source:- N. Y. Journal of Commerce) 
(Pigures in millions of dollars) 


3 Apr May Jun Jul Aug Sep l 
988 601 676 282 580 490 
12.40 7,55 8.60 3.54 7.29 6.16 6.51 14.62 7.78 | 


Fig. 223. The Seasonal Cycle Computed from One Cyclic Period. 


~ cycle. If we wish, we can calculate the various months in 
this cycle as related to the total for the cycle, that is, change 
each month into a percentage of the total for the year, the 
latter being 100%. 

The trouble with this crude use of a single year as an index 
or indicator of cyclic fluctuation 18 two-fold. For one thing 
it does not take any account of secular trend—which in the 
case of a young and rapidly growing business will be very 
marked—and as a result December sales may appear to be 
seasonally larger than January sales, though in fact they are 
really smaller, because every year the following January sales 
exceed the last December sales, just as within the calendar 
year the last December sales always exceeded the last January 
sales; the result of this error is to skew the seasonal fluctua- 
tion curve around in the cycle, tilting up one end of it, giving 
us a warped picture of the cyclic fluctuation, in which the 
warping or tilting may be so great as actually to shift the 
location of the peaks and valleys. If the data be the record 
of sales by an individual concrn, no matter how true a picture 
it may afford of the experience of the company, it does not 
give a true picture of the changes in the market, the seasonal 
variations in consumer demand. 


248 CHARTS AND GRAPHS 


The simplest way of correcting for the secular trend or 
general movement of the phenomenon is to take the months 


NEW INCORPORATIONS 
Capital Invested in New Enterprises 
Mose authorized capital equaled or exceeded $100,000 
Princ ctpey oaeee U. Ss. 


(Source:- N.Y. eeu of Commerce) 
(Figures in millions of dollars) 


Jan Peb | Mor Apr 


1921 Amount 1,243 654 355 988 
13,962 | 13,457 13,036 | 12,670 
8,92 4.86 133 7.81 
12,25 6.77 10.19 10.87 


676 
Moving Totel 11,206 
6.03 


6.39 


Percentage 


Corrected 


Fig. 224. The Seasonal Computed from the Trend. 


not as percentages of the total for the calendar or fiscal year 
of fixed span, but to take them as percentages of the moving 
total (“for the current twelve months’’), for the same months. 
The results will no longer add up to 100%, but will be less 
than 100% if the moving total has fallen and more if it has 
risen. The monthly percentages of the moving totals must 
therefore be summed up and corrected so that their sum 
equals 100% (by dividing them by their sum). The result of 
this process may be taken as in most cases an entirely satis- 
factory record of the typical seasonal fluctuation, during one 
year. 


JAN. FEB. MAK APR. MAY JUN JUL. AUG. SER OCT. NOV._ DEC. 


Fig. 225. A Remarkable Case of Changing Cycle Fluctuations. 


Typical seasonal changes in interest rates on 60 to 90 day commercial paper for 
the years 1890 to 1908 and the years 1917 to 1921. Weekly variations are shown 


as percentage deviations from the annual average. —Permission of Mr. Carl 
Snyder. 


os me GLCLES 249 


The second objection to the method still remains, however. 
This objection is that the cycle is estimated upon a single 
year’s experience only. The cycle shown by another year 
might be somewhat different. And how do we know that 
one year is any more representative of actual conditions than 
another. Of course, in the case of businesses (or other phe- 
nomena) effected by the war (and this includes most business 
and economic records), we might quickly throw out the war- 
time record, as being wholly unreliable. But in the absence 
of special reasons for discarding certain periods or records as 
unrepresentative, we may be confronted with many years of 
equal significance which yield different cyclic curves. And in 
such cases it would be wrong to trust one entirely and discrim- 
inate against the rest. The obvious thing to do is to calculate 
the seasonals for each of these years, by the method above de- 
scribed, and then average them together, to get an average 
seasonal. The resulting curve would meet the second objec- 
tion and be representative of the entire experience, for which 
records are available. 

Perhaps the most important use of seasonals in ordinary 
business statistics, is the calculation of “quotas,” or planned 
“schedules” for the future. In a sales department, for ex- 
ample, the quotas assigned in advance to the salesman, or to 
the sales districts, should be as fair as possible, and to assure 
this, the typical seasonal fluctuations should be known. Our 
problem then becomes slightly different. We no longer want 
the seasonals most typical of the entire experience of the past, 
but we want the seasonals which may be considered most 
typical of the immediate future. A simple average of the 
seasonals for many years past would give too little importance, 
perhaps, to recent developments. It may be that, through 
advertising, or through changes in market conditions, the con- 
sumer demand has been shifted about in the year (usually to 
become more level, that is, regular). For such developments 
it is plain that the later years are more truly representative 
than the earlier ones. 

In the calculation of quotas, therefore, it is well to “weight” 
the later years more heavily than the earlier ones before av- 
eraging. Ordinarily it is satisfactory to weight each year twice 
as heavily as the preceding year. Other weighting systems 
can be used, but this particular arrangement leads to a most 
easily calculated average seasonal which has been devised and 


250 CHARTS AND GRAPHS 


used by the author for a long time under the convenient, though 
somewhat loose, name of the ‘compounded average.’ It has 
the advantage of being easily carried on from year to year 
without extensive re-calculations, an important factor in a 
busy office, and it also avoids all question of how many back 
years to include, by making all except the last four or five 


NEW INCORPORATIONS 
Capital Invested in New Enterprise 
Those piteonises capital equaled or praasned $100,000 
eda States, U. S. 
1918-1921 
(Source:- N. Y. Journal of Commerce) 
(Figures in millions of dollars) 


Feb ai Mer Apr may 2 dul 


A Se Oct 
‘® Ug P sail 

4,12 4.69 6.35 8.05 6,23 5.57 4.77 7,24 4.91 
5.98 6.80 9,21 11.70 9.04 8,08 6.92 us) 7.13 


68.90 
100.00 


1918 | Percentage 


Corrected 


le 
1919 Percentage 18,87 11.79 12.71 16,22 20,68 27,05 24,15 12,63 23.49 22,44 209.88 
Corrected 2,00 5.62 6.08 7.74 9.86 12,80 11,50 5.98 11.20 10.71 100,00 
votal | 1918 ond 1919 18.12 11,60 12,88 16,95 21,56 21.84 19,58 12.90 21,71 17,84 200,00 
Average 9,06 5,80 6,44 8.46 10.78 10.92 9,79 6.45 10.61 8,92 100.00 

im a5 
1920 Percentage 15.76 7.57 | 8.44 7.90 | 7.97 7.41 7.41 5.27 | 8,64 7.54 92.24 


Corrected 17,10 8.21 9,16 8.57 8.62 8.04 7671 6.71 6.11 8.17 100,00 


‘otal | 1920 and Average 26.16 14.01 | 15.60 | 17.03 19,40 18.96 17,50 12,16 16,92 17.11 200,00 


Compound Average 13,08 7,00 7.80 8.52 9.70 9.48 8,75 6.08 8.46 28.55 100,00 
apie: — 


1921 Percentage 8.92 4.86 | 7.33 7.81 5,07 6.03 2.75 5.68 §.21 §.76 71.89 


Corrected 12.25 6.77 10.19 | 10.87 7.06 8.39 3.93 8.19 7.26 8.02 100,00 


Total | 1921 and Average 25.33 13,77 17,99 19.39 16.76 17,87 12.68 14,27 15.72 16.57 
2.66 6.89 9,00 9.69 8.38 8.93 6.34 7.14 7,86 8,29 
a Ieee 


200,00 
100.00 
=<) 


Compound Average 


Fig. 226. The ‘‘Compounded Average’’ Seasonal. 


negligible. The trick is to average the seasonals for the first 
two years (which can be done at sight), and than average the 
resulting average with the next year to get a new average, 
continuing this process through the years and always working 
by inspection. 

With the method here outlined to use when you wish to 
ascertain the seasonal fluctuations in your data, using the 


6 Of course, in this “compounded average” the weighting is not two to one for the 
first two years, but with the exception of the first year the weighting is in this ratio 
throughout, and in a very few years -the importance of the first year is rendered so 
negligible as to be lost. 

By other weighting systems, it is meant that ratios of three to one, or of one to 
two-thirds, or of one to three-fourths, or the like, can also be easily used and currently 
maintained (that is, brought up to date) almost by inspection. 

The theory of the compounded average is very simple, and appears to be sounder 
than that of any fixed average seasonal. It is believed to be an original contribution 
to the science of averages, which should have particular value in economic work with 
phenomena undergoing changes in seasonal fluctuations. A very spectacular case of 
such a phenomenon was the behavior of the interest-rates for loans in New York 
after the establishment of the Federal reserve system, when a previously marked 
seasonal was almost entirely wiped out in a few years. The compounded average 
affords a sort of moving or progressive seasonal well adapted to such cases. And, in 


the ease with which it is brought up to date, it is, mechanistically, a decided labor- 
saver and time-saver. 


CYCLES 251 


“compounded average” in the place of the simple average for 
quota-making, and with the moving total? and average previ- 
ously described for the elimination of the seasonal when you 
wish the real underlying movement, loosely called the secular 
trend, in your data, you are equipped with the mathematical 
means necessary for the successful use of the following charts. 
Apart from the need of the cyclic change in quota-making, 
the usual need is for the secular. trend and the latter is indeed 
useful not merely for the following charts, but for a wide vari- 
ety of purposes in statistical work. It is therefore the more 
important trick to have up your sleeve, in attacking either 
business or sociological statistics. However far it may fall 
short of a true secular trend, it still gives a significant and 
easily understood smoothed curve. Though still little known 
to the average executive, it is proving extremely popular among 
those who use it, and has been credited by some business sta- 
tisticians, chiefly those who use the device described in the 
next chapter, as being the only part of the data in which the 
executive should be interested. 

7 By an oversight, the tables in the discussion of the compounded average all 


show the months as percentages of the moving total for “‘previous” 12 months. It 
is obvious that the moving total of “‘current”’ 12 months should have been used. 


CuapTer XXII 
ZEE-CHARTS 


In this chapter we enter the accountant’s paradise, and in- 
stead of simplifying our data and presentation, we multiply it 
three-fold, by adding to each original series of data, its cumu- 
lative and its moving total series. The result is a chart which 

‘shows simply and coherently everything about the data which 
can be shown. It takes much space, for each important 
period (i.e. year or month) of data should be given a separate 
chart, and its use is therefore better restricted to a few series, 
whose importance is sufficient to justify their treatment in 
this thorough and painstaking way. In its way, this chart is 
the last word in the analysis of and research into past his- 
torical data.! 

The “Zee-chart” gained its name from the fact that its 
three curves roughly form the letter ‘“‘Z.”’ These three curves 
are, first, the curve of the original data, second, the cumula- 
tive curve, and third, the moving-total curve. In common 
practice there are three kinds of these charts, depending upon 
the time period of the original data. Where the original data 
consists of monthly figures, the chart shows twelve of these 
figures to form one year; when the data is weekly, fifty-two 
weekly figures are combined in one chart to show one year; 
and when the data is daily, thirty or thirty-one days are com- 
bined to show in one chart a month. In the first two cases 
the moving-total is an annual one and in the last case it is a 


1 The Zee-chart is rumored to be of German origin, but appears to have had a 
somewhat later independent American discovery. It has received its greatest develop- 
ment at the hands of Mr. Willard C. Brinton, consulting engineer and author of 
Graphic Methods for Presenting Facts. The present writer is informed that the combi- 
nation of monthly and moving total curves on standard scale combinations was worked 
out by Mr. T. R. Robinson, and the addition of the cumulative curve was suggested 
by Mr. Wallace Clark. Of late, the Zee-chart has been further modified in its form 
by Mr. Arthur R. Burnet who has also suggested the omission of scale-figures to 
focus attention upon the curves, and has invented scale-finding machinery to facili- 
tate plotting, 


252 


ZEE-CH ARTS 253 


monthly one, though it is to be noted that in most businesses 
a monthly moving-total has little significance. The Zee-charts 
can, however, be made up of any combination of time units 
desired. The most practicable one, and the one which will be 
herein described, is the Zee-chart of monthly data, twelve 
months comprising one chart. Its ready adaptability to the 
needs of the business man and accountant makes it extremely 
useful for recording data in these fields. 


Defects Losse $ 1920 


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Authorized RWB. 3/14/18 
Formula Based on reduced sales value, 


Data Available Tueeday 


Source 


Comparisons 


8 


Permission of the Ronald Press. 
Fig. 227. A Year by Weeks. 
Showing the arrangement and methods advocated by Mr. Burnet. 


By this time, if you have understood the last two chapters, 
you will be protesting that it is not practicable to show a 
moving total curve on the same chart with the curve of the 
original series. Why? Because the moving total for twelve 
months is twelve times as great as the average monthly items, 
and if the moving total curve is to be shown, the curve of 
monthly figures will lie very close to the bottom of the chart, 


696" 
Tos‘a 
62L'8 


Journal of Commerce) 


1916-1921 


NEW INCORPORATIONS 
{ Capital Invested in New Enterprises 
) Whose authorized capital equaled or exceeded $100,000 
Ye 


Principal States, U. S,. 
(Figures in millions of dollars) 


(Source:- Ne 


eS 


j 


9066 6969 
298'6 = 6146'S 
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90e'TT LTT'S 
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=a 


In the originals the annual moving 


Four Zee-Charts Forming a Single Series. 
Notice the different scales for monthly and annual data and the corresponding positions of the data captions. 


total and cumulative data and curves and scale are in red. 


Fig. 228. 


1918 


ZEE-CHARTS aoe 


its fluctuations hardly visible. The objection is well taken, 
but the Zee-chart cleverly dodges this difficulty. How? By 
the simple device of using two scales. A large scale is used 
for the original monthly figures, but a small scale is used for 
the cumulative and moving total curves, whereby they are 
brought down on the chart to not more than two or three 
times the height of the monthly curve. The ratio between 
these two scales has been more or less standardized, the 
monthly-curve scale being five times as great as the annual 
cumulative and moving total scale. When the data are weekly 
and the moving total is a 52-weeks total, the ratio is larger, 
the weekly scale being twenty times as great as the cumulative 
and moving total scale. When the data is daily and the moving 
total monthly, the ratio is again different, the daily scale being 
ten times as great as the monthly cumulative and moving 
total scale? 

Standardized practice also has it that a color distinction 
should be observed between the two scales and their corre- 
sponding curves and data. You should enter the individual 
series in the data at the top of the chart in black, plot the 
curve therefore in black and enter the scale itself in black 
lettering. But the cumulative and moving-total figures should 
be entered in red in the data at the top of the chart, likewise 
their curves should be drawn in red ink and their common 
scale entered in red letters. This distinction is an excellent 
one as it causes the curve of original data to stand out most 
prominently while the moving total curve or trend is perfectly 
clear, some distance higher on the chart. 

As you have seen in an earlier chapter, the moving total 
of a series can be plotted at any point within its period, and 
can be a moving total either of the current twelve months, of 
the following twelve months, or of the past twelve months. 
It is the last kind of moving total alone which is used in the 
Zee-charts. From this fact, two advantages arise. In the 
first place all three curves are entered at the same time on the 
same ordinate and a hasty reader who has no time to analyze 
the figures, gains no false impression that the chart is not 


2 A very shrewd suggestion has been made by Mr. Arthur R. Burnet, consulting 
statistician and graphic expert, that the scale-figures be omitted from charts which 
are to be shown to the non-technical business man, in order that his attention may 
not be diverted from the behavior of the curves. The fact that data is always in- 
cluded in the Zee-chart, and hence scales can always be ascertained though not cali- 
brated, gives to this suggestion unusual merit. 


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TVLOL TVANNY 


CHARTS AND GRAPHS 


ZEE-CHARTS Pas] 


thoroughly up-to-date, as he is likely to when the moving 
total is plotted upon an earlier ordinate. In the second place, 
the cumulative and moving total curves always come together 
at the end of each chart by this device, since it is obvious that 
the cumulative for twelve months is the same as the total for 
the past twelve months. This coincidence of the two curves 
is an important element in the simplicity of the chart and also 
automatically checks both computing and plotting. 

The Zee-chart should be prepared upon the same type of 
paper as other historical curves, that is, its field should be 
positioned in the lower righthand corner of the page, close 
to the edge of the paper. For Zee-charts are generally pre- 
pared in sets for data extending over many years and requiring 
as many charts in the set as there are years. It is important, 
therefore, that the reader be able to fan the charts out, each 
overlapping the next so as to afford a general view of the entire 
series of years in a small space. The narrow margin to the 
right of the chart on each sheet produces gaps between the 


Current 


6 


6 7 
MOVING ANNU, 
me | Bee TOTAL 


Zen 
oo” 


[cP 
PTT 


o 
1° 


1 
6 ? @ . 1 


Permission of Mr. Arthur R. Burnet and the Ronald Press. 
Fig. 230. Two Charts Fanned Upward to Study Seasonals. 


258 CHARTS AND GRAPHS 


charts when laid out in this way but these gaps are of benefit, 
forming slight breaks between years without destroying the 
general continuity of the set of charts. The low position of 
the field upon the paper facilitates the study of seasonal fluc- 
tuations by overlapping charts one above the other. Needless 
to say, the scale on all charts belonging to a set, should be 
uniform and rigidly maintained within the set in order that 
the curve running through the charts may be homogeneous. 

The scale on a Zee-chart should be somewhat smaller than 
on most historical curves, for the reason that you probably 
expect to continue the Zee-chart series in the future, and you 
must allow ample margin for future growth of the business 
and future peaks which will rise higher than the past peaks. 
To find the scale for a Zee-chart, therefore, look back over the 


series of moving-total figures in your data and locate the 


highest peak figure in the past, and then select a scale which 


Observed CURRENT MONTHLY SERIES CUMUL‘ TIVE ANL MOVING TOTAL SERIES 
Peak in : 


Type of 
Entire Field Scale- 
Moving to use figure Valve Rule to Pletting i Value Rule to Plotting 
Total for the first of Data use in Velue of © of Lata use in Value of 
Series Chart Abecissa to Flot Plotting Rule-unit Abscissa to Plot Plotting Rule-unit 


Fig. 231. Table for Zee-chart Scales. 
For standard chart-fields six inches high. 


will bring this figure about half way up the chart. Selection 
of the scale is perhaps one of the most difficult parts of Zee- 
chart making, and it is convenient to use a table of scales 
similar to the table for historical scales given in a previous 
chapter, modified only by increasing the key numbers one- 
third to lower the peaks to half way up the chart. 

One of the most interesting features of Zee-charts is their 
use in what is called “light analysis.” Light analysis is a 
method of comparing two curves or charts drawn on similar 
scales. You place the two charts together, one upon the 


ZEE-CHARTS 259 


other, and hold both up to the light. Both curves will then be 
visible, enabling you to compare them minutely. In some offices 
where a large number of Zee-charts are used, a machine called 
a “light box” is kept for this work. The light box consists of 
an electric light underneath a. piece of ground glass, which 
throws. a uniform illumination up to the paper laid upon it. 


The charts should therefore be made on highly translucent’ 


paper, and the field should be positioned with absolute uni- 
formity on every sheet. These are considerations which hold 
not merely for Zee-charts but for all curve-charts to be sub- 
jected to light analysis. 

A number of attempts have been made to adopt the Zee- 
chart to current operation control. It is easily seen that by 
extending or projecting the cumulative curve on any up-to- 
date chart in which the end of its period (e:g. year) has not 
yet been reached, an estimate can be quickly made of the 
probable amount to which the future months will bring the 
entire year’s total. Moreover, if a quota for the year has been 
made in advance, the cumulative for this quota can be easily 


- plotted in pencil or, best of all, in yellow ink and a comparison 


between the red cumulative for actual performance with the 
yellow quota cumulative will show how well or poorly the 
quota is being fulfilled. A slightly different method is that of 
showing the quota cumulative as a straight sloping line, and 
entering the curve figures and SsiTAH SC as percentages of 
the amount which would have been necessary to meet this 
quota. But these various methods have not been so successful 
as to find general acceptance, as they tend to fill up the chart 
with too much detail. 

The plain fact is that the Zee-chart is a “looking-backward”’ 
chart. The best that can be said of it is that it is an ideal 
method of historical research. The sales of your chief line or 
department should be plotted in this way, in order that you 
may study their past history to the best advantage. The chart 
is designed to show you at once the individual monthly or 
periodic fluctuation, their general trend or moving total, and 
the cumulative or total to date for each year, and from Tes 
three accounting elements for each year, you can see just when 
sales began to fall off or rise, what the seasonal fluctuation 
was, and how each year’s individual progress compared with 


®The paper may be highly translucent without being in the least transparent. 


260 CHARTS AND GRAPHS 


Raw 'atertal- Receipte 


Movina 


Cumulative Annual 
Total 
102 630 


9 642 


3/25/15 


Source Purchasing Dept. Storeroom 


Market prices 
Ae» Purchase on new contract began 


Leos returns 


Data Available 10th 


e 
2 
° 
& 
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Comparisons Total Inventory, Ordere Rec, ,Sales, 


Formula Net equals gross lese shrinkage 


Authorized E,J.C, 


Notes 


Permission of the Ronald Press. : 
Fig. 232. Mr. Burnet’s Arrangement. 


that of other years. The Zee-chart is, among amount-of- 
change charts, the last word in historical research and detail.4 


4 An excellent description of the Zee-chart is to be found in a series of articles by 
Mr. Arthur R. Burnet in Management Engineering, beginning Sept. 1921, pp. 153-160. 

The first American description is that by Mr. John Wenzel in an early issue of 
the Scientific American Magazine. 

The best adaptation for forecasting and quota-measuring has been made by 
Mr. John Scoville, Maxwell Motors, Detroit. 


CHAPTER XXIII 
PROGRESS CHARTS 


From the paradise of the accountant and historian let 
us step into the paradise of the operating executive. The 
operating executive is the man who sees to it that things are 
done. His interest begins and ends with the job to be done, 
how much of it has actually been done, and how much remains 
to be done. When told that a job has not yet been finished, 
he is not interested in excuses and reasons, he is not interested © 
in why or when the performance fell below the schedule, he is 
only interested in the amount remaining to be done and the 
job of getting it done. We shall not expect to find him satisfied 
with a retrospective or “looking-backward” chart. For him, 
the “looking-forward” chart! 

Far and away the most “forward-looking” chart known is 
the “progress” chart. It is the product of the man who was 
probably the greatest engineer America has ever produced, 
the late Mr. H. Le Gantt! It is significant that this chart 
method was devised by an. engineering type of mind, for it is 
admirably adapted to an executive control of operation. 
Compared with its dynamic influence on the actual control of 
operations, nearly all other types of charts seem to justify 
the assumption that the word “‘statistical’’ is derived from the 
word “‘static.” 

1 Henry Laurence Gantt was born in Maryland on May 20, 1861. He received 
the degree of bachelor of arts in 1880 from Johns Hopkins University and in 1884 
the degree of mechanical engineer from Stevens Institute of Technology. He died 
November 23, 1919, at his home in Montclair, N. J. 

Mr. Gantt was associated with Frederick W. Taylor in his early work at the Midvale 
and Bethlehem Steel Companies and a few years later established his own consulting 
practice as an industrial engineer, which he carried on until his death. 

Among his clients were many of the most advanced manufacturing companies of 
this country. During the war he devoted his entire time as well as that of his staff 
to the solution of the Government’s problems of production and management. He 
acted in a consulting capacity for the Ordnance Depaftment, the Naval Aircraft, 


the Shipping Board, and the Emergency Fleet Corporation. 
Mr. Gantt developed a method of paying workmen according to the results they 


261 


204 — CHARTS: AND GRATIS 


The Gantt progress-chart presupposes a definite detailed 
schedule or plan made out in advance and generally called the 
“quota.” The chart itself merely measures the subsequent 
actual performance, when it takes place, against this pre- 
determined schedule or quota, and shows emphatically 
whether or not this quota is being met. The chart shows 
incidentally how much of each past month’s quota has been 
accomplished, but primarily it shows how much of the cumu- 
lative or total quota to date has been accomplished. The chart 
never shows trend or moving total, nor does it necessarily 
show even the individual monthly figures; its main function is 
to show how much of the schedule has been performed up-to- 
date and how much remains to be done. 

As we might have expected in a chart for an: operating 
executive, the progress-chart compresses its information into 
very small space. Where the Zee-chart expanded every series 
of figures three-fold and used a separate sheet of paper for 
each series, the progress-chart combines twenty, thirty, or 
even more, series upon a single page. An entire business or 
industry can be summarized upon one of these charts, each 
of its thirty or more items being in turn shown in detail with 
thirty or more sub-divisions on subordinate charts. In the 
course of time, the Gantt progress-chart will come to be 
recognized as the sine qua non of management, whether it be 
sales management, office management, or production and 
factory management. 

Strictly speaking, the Gantt progress-chart is not a curve- 
chart at all. It is rather a horizontal bar-chart, very peculiarly 
constructed. But the relation between bar-charts and amount- 
of-change curve-charts is so intimate that it can best be 
examined here. If you like to so consider it, the Gantt 
progress-chart 1s a combination of many curve-charts, each 
flattened out into one dimension and all placed close together 


accomplished, which is known as the Gantt Task and Bonus; he developed the theory 
that the cost of an article includes only those expenses actually incurred in the pro- 
duction of that article, and that the expense of maintaining one machine in idleness 
can not be charged into the cost of the output of another machine. In accordance 
with this theory, he worked out a method of arriving at costs of idleness and of work; 
he originated the Gantt Chart, which compares the amount of work done in a given 
time with what should be done and emphasizes the reasons for failure to attain that 
standard; he introduced a change in the installation of management methods from the 
old type, which organized from the top down, to a new type which builds from the 
bottom up. Work, Wages and Profits” (1910), “Industrial Leadership” (1916), and 
“Organizing for Work” (1919), are the titles of the most important books written 
by Mr. Gantt.—Wallace Clark. 


- 
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PROGRESS CHARTS 263 


on a single chart. Or if you prefer, the Gantt progress-chart 
is a series of horizontal bar-charts with cumulative bar-charts 
superimposed. 

The scale of the progress-chart is one of its most interesting 
features. At first glance, there appear to be as many scales 
on the progress-chart as there are bars, or items. That is to 


say, every bar on the progress-chart appears to have its own 


scale or set of values for the horizontal distances through 
which it passes. And, unlike all other charts, the horizontal 
distances appear to have no equal and uniform values through- 
out the length even of a single scale; on the contrary, the values 
given to equal horizontal distances, or spaces between vertical 
lines, seem to change weirdly from space to space throughout 
each individual line. But the secret of the puzzle is very 
simple. The common and proper scale for progress-charts is 
“time.”’ Each equal distance represents an equal unit of time, 
and you will find a time scale placed at the top of each progress- 
chart, one single uniform scale for the entire chart. Time, then, 
is the measure or unit of measurement against which perform- 
ance is measured. In Mr. Gantt’s words, “time is the one 
common thread running through all operations,” and time is 
therefore the one basis on which all performance should be 
judged by executives. 

Now it is obvious that the number of dollars worth of 
goods sold during a month cannot be taken directly as a part 
of, or measured directly in terms ‘of, the number of days in 
the month. The amount of sales, production, or other per- 
formance, and the length of time involved in the performance 
are numerically incommensurable quantities. We must there- 
fore have a ratio or co-efficient between the two, that is, we 
must make up our minds that a certain unit of time is to equal 
a certain volume of sales or other performance. Then the 
actual amount of performance can be judged in terms of this 
predetermined quantity, which has been decided upon for the 
given period of time. And these predetermined quantities are 
the items which at first glance appeared to form the irregular 
scales for each bar. 

In short, the progress-chart measures actual performance 
in terms of a standard. This standard is shown by the small 
figures in each space and is graphically represented by the 
entire space itself. Actual accomplishment is graphically 
recorded by a bar drawn across the part of the space which 


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PROGRESS CHARTS 265 


corresponds to the percentage (of the standard), which has 
been accomplished. In other words, the standard for each 
space (or period of time) is considered to be 100% for that 
period of time, and the actual accomplishment during that 
period of time is taken as percentage of this standard, and 
shown by a 100% bar, in which the shaded portion represents 
the part accomplished, the unshaded portion the part not 
accomplished. ‘This is literally true of the light lines or bars 
which begin afresh at the beginning of each new space and 
indicate the monthly or periodic performance or accomplish- 
ment. The method is not quite so simple, however, in the 
case of the cumulative performance or accomplishment, that 
is, the accomplishment from the beginning of the entire 
period shown on the chart, to date. This cumulative of per- 
formance is shown by heavy bars. Here (1) all the standard 
cumulatives which are less than the accomplishment cumu- 
latives are considered wholly performed and 100% done, 
and the heavy cumulative bar is drawn entirely across them; 
(2) the remainder of the accomplishment cumulative, after 
subtracting the last standard cumulative, is taken as a per- 
centage of the next individual period standard and (3) the 
cumulative bar is drawn correspondingly across the corre- 
sponding percentage of the last period space. 

A simple example will make this clear. Suppose we are 
allowed by the publishers of this book ten months in which 


6 
PROPOSED 
SCHEDULE 


Month Quota 


1 re) 

2 1 

3 4 

4 6 

5 6 

6 6 

q 6 

8 6 

9 6 
10 On 

Total 50 
Fig. 234. 


to prepare it. As a matter of fact, the book is the result of 
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PROGRESS CHARTS 267 


methodical sort of person, we sit down and prepare an outline 
of the book, and discover that it will take about fifty chapters 
in which to tell all that you should know about the subject. 
We then prepare a schedule showing how long it will take us 
to write each chapter, and decide that we can write the first 
three chapetrs in the first month, one chapter in the second, 
then four and thereafter six chapters a month. Next we pre- 
pare a progress-chart of this work showing ten months on the 
chart, one space for each month. We enter the number of 
chapters to be done each month in the upper left hand corner 
of the space for each month. We also enter the cumulative in 
the upper right hand corners of each space, showing that at 
the end of the second month we will have written four chapters, 
by the third month eight chapters, by the fourth, fourteen, 
and so on, until at the end of the tenth month fifty chapters 
are written. This checks our monthly schedule. 

‘Time passes and we are writing, patient reader—though you 
might not guess it—we are writing with meticulous and pains- 
taking care, If at the end of the first month, only one and one- 


Progress Chart 


Fig. 236. The Chart on Jan. 31st. 


The light line records a performance of 50% of the month’s quota; the heavy line 
records 50% of the first month’s quota cumulative. The V-shaped mark shows 
date of last entry. 


half chapters have been completed halfway across the first 
space, when we should have written three chapters, we draw 
two lines, one light and one heavy, the light one being above 
the heavy one. By this we know that at the end of the first 
month we have only done 50% of the month’s quota. In the 
second month we finish the second and also a third chapter. 
A new, light or monthly bar can be drawn all the way across 
the second space and a second light bar halfway across, above 
it, showing that we had done our bit and 50% more in the 
pecoad month. But as we were short in the first month’s 
work we are still short to date, having done three chapters 


268 CHARTS AND GRAPHS 


when we should have done four. These three chapters finish 
our quota for the first month only and leave nothing accom- 
plished out of the second month’s quota. We therefore draw 
the heavy cumulative bar completely across the first space 


arash aedsna Chart 


= ETE 

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TTT 

Fig. 237. The Chart on Feb. 28th. 


The light lines show 150% of the second month’s quota performed in the second 
month; the heavy bar shows total performance to date (Feb. 28) just one month 
behind schedule. 


but not into the second space at all. A glance at the chart 
now shows us that we are short one month’s work (though we 
have exceeded our quota in the second month). If in the 
third month, in an excess of energy, we write six more chapters, 
we shall have done 150% of the third month’s quota of four 
chapters and shall therefore draw a light line all the way across 
the space, and over it another light line one half of the way 
across the space. But the cumulative or total to date will be 
nine, enabling us to draw the heavy bar all the way across the 
page to the end of the third month and one-sixth of the way 
across the fourth space, since the fourth months’ quota was six 
chapters, of which we have done one. A glance at the chart 
now shows us that we are ahead of schedule. ‘This illustration 
may seem wholly personal, but the sales or factory manager 
who cannot see in it a method applicable to his own problems 
is not worth his salt. 

You will now understand something of the unique merit 
of the progress-chart. It has an uncanny power of making 
human judgment. It does not merely record what has been 
done, but in addition thereto it records whether this accom- 
plishment has been good, bad, or indifferent. It does not 
merely put the question, but it also gives the answer. It 
weighs every fact in the balance and states in unmistakable 
terms the judgment. If you have fallen down on your job, 
the chart does not waste emphasis on why, when, or how you 
fell down, but places before you in a way which you cannot 
dodge the fact that you have fallen down. It is of course 


269 


PROGRESS CHARTS 


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you will) the chart will bear emphatic witness to that fact; it 
will proclaim your success from the housetops.2 

Picture to yourself a busy sales or factory manager receiv- 
ing the usual detailed report on the production or activities of 
his various departments. ‘The record is in tabular form show- 
ing what each department has performed during the month, 
or during the year to date. Before he can decide whether 
the work is satisfactory, he must study each figure, and in 
the light of his knowledge of all the various factors and cir- 
cumstances, decide whether each item shown is satisfactory 
or not. It is a task which will cost him many hours of close 
concentration on every occasion when the report is submitted 
to him, and each time he will find difficulty in remembering 
just what were his previous decisions about each item. An 
executive’s time is being taken up, not in getting things done,. 
but in thinking about them. It would not be so bad if it 
could be accomplished once for all; the pity of it is that it has 
to be repeated every time a report is submitted to him. 

Now let us help this executive by giving him a Gantt 
progress-chart at the beginning of the year or period for which 
the chart is to run. Let us sit down with him and ask him, 
once for all, to consider the various factors which in his judg- 
ment will be the basis of satisfactory work during the year to 
come, and in the light of those factors to determine upon a 
reasonable standard for the coming year’s activity. Often he 
will give us merely the salient factors and their approximate 
influence and leave to us the working out of a detailed schedule 
in accordance with them. Often we will get much of the 
detail from subordinates closely in touch with them. In any 
case, what we are going to try to do is to devise for the entire 
coming period a schedule of reasonable expectation of the 
business for the coming period (e.g. year) worked out in detail 
for each element, department, or other subdivision, and for each 
unit of time (month, week, or day). This reasonable standard 
or schedule of expectation is sometimes called a “quota” or 
“task.”’. It will often work very well as a quota, or basis of 
rating by merits or demerits, subject, of course, to unforeseen 


2 “Unlike statistical diagrams, curve records, and similar static forms of presenting 
facts of the past (Gantt) charts . . . are kinetic, moving, and project through 
time the integral elements of service rendered in the past toward the goal in the 
future.’—Walter N. Polakov, Principles of Industrial Philosophy, Proceedings of the 
American Society of Mechanical Engineers, December, 1920. 


CHARTS AND GRAPHS 


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reached what the 


changes in the various factors or circumstances. 
1s not important, and whether we call this a 
or a schedule, the point is that we have 


PROGRESS CHARTS 273 


executive considers to be the best basis for the judgment of 
work done. And we have finished the major part of the work 
of preparing a Gantt progress-chart. We enter these figures in 
the blank form and wait for performance to show the accom- 
plishments as they occur, graphically on the chart. 

Sometimes the difficulty of preparing a reasonable quota is 
so great, or the factors and circumstances which will be in- 
volved in the work are still so obscure, that a quota is not 
desirable. Nevertheless, you will find the executive later on 
using some figures or other for comparison. Most frequently 
he will be using the figures for the previous year as a basis of 
comparison. So when we cannot make up an ideal standard, 
we merely use the last year’s record, or perhaps an average 
of several recent years, in the place of a quota on the chart. 
Gantt engineers, working on production and office problems, 
have developed a scientific technique in quota or schedule- 
making which is intimately connected with and greatly simpli- 
fies the problem of cost accounting in the plant. But whether 
these scientifically reliable quotas, or merely rough guesses, or 
simply the previous records, be used as the standard, it is all 
one to the progress-chart. The chart will use any basis 
adopted, and judge the performance in terms thereof. 

Note the time-and-labor-saving value of the progress- 
chart. After the fundamental decision as to schedule standards 
have been made, the chart works automatically. It is a ma- 
chine, passing up to the executive his own judgments. All the 
labor involved has been transferred to clerks. The executive 
merely glances down the chart, noting the length of the heavy 
bars. His attention is immediately drawn to the exceptional 
performances which he would wish to study. There is no 
dodging or forgetting these exceptional cases as shown on the 
chart, and the executive is enabled either to discount the ex- 
ceptional cases in the light of further developments or unfore- 
seen circumstances, or to take immediate action where such 
action 1s called for. 

This method of charting is so simple that Gantt engineers 
are accustomed to install it in shops for the use of foremen, 
as well as for the central planning and executive departments. 
They are accustomed to enter the quota in ink and the graphic 
bars in pencil (black lead pencil). The maintenance of these 
charts requires: no special staff of draftsmen or computers; 
they are used and filled in by the workmen and foremen them- 


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CHARTS AND: GRAPHS 


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274 


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PROGRESS CHARTS ae 


selves, the bars being roughly sketched in without regard to 
nice appearance. Where the charts are used in a statistical 
department, or for the higher officials of a concern, it is perhaps 
better to observe more care in the appearance of the chart, 
and in this case colors can be used to advantage. The monthly 
or individual time unit quotas can be typed on the chart in 
black typewriting and the cumulatives in red; the light bars can 
be ruled in black India ink with a drawing pen, the heavy 
cumulative, in red waterproof ink. If this color distinction is 
observed, the ends of the red bars each month should be marked 
with light black cross-lines vertically cutting the red bar into 
segments, and showing the position of the cumulative at vari- 
ous times in the past. The initial letter in the name of the 
month can also be entered in black on the red bar to identify 
these points. When no color distinction is made and the 
cumulative bar is black, it is better to draw slight notches 
below each final position of the cumulative bar to keep this 
record of cumulatives. The first line upon a progress-chart is 
usually used for the total of the other lines and its bars are 
drawn with extra wide or heavy rulings, for the sake of em- 
phasis. 

A further refinement has been adopted by Gantt engineers 
in the binding of these charts. The charts are bound at the 
righthand edge of the paper in order that the stubs at the left- 
hand side may be quickly seen. This is of benefit where a large 
number of progress-charts are kept in bound volumes. 


ee 


Fig. 242, Data on a Short Fly-sheet—the Gantt Way, 


jae CHARTS AND GRAPHS 


It has been said that the data should always be available 
with every chart and the Gantt progress-chart is no exception. 
Performance or accomplishment is portrayed graphically by 
the bars, but the figures which these bars represent are not 
shown on the chart. A blank form, similar in its ruling to the 
form of the chart, is laid immediately above the chart page 
with the column for stubs removed, so that the page will be 
short and the stubs as entered on the chart will be visible for | 
both chart and data sheet. On this blank form the data of 
performance is recorded in writing, the figures for both indi- 
vidual time units and cumulatives being shown in the same 
spaces as used for their corresponding quotas on the chart. 
Because this method presents the data rather than the chart 
itself when the page is first opened, and the data sheet has to 
be turned over to read the chart-sheet, the writer’s individual 
practice is to reverse the arrangement and place the chart on 
the short sheet and the data on the full sheet underneath— 
an act of heresy which Gantt engineers are not expected to 
endorse. 

It is with reluctance that we leave the subject of progress 
charts. So far as the executive interested in the graphic con-. 
trol of operations is concerned, the entire subject of charts and 
graphs begins and ends with this chapter. No other method 
of charting has yet been invented which presents the essen- 
tials of operating so forcibly, so clearly or in such small space. 

The details for each department can be shown on depart- 
mental sheets with the total for the department at the top 
of each chart. The various department totals can be brought 
together upon a single plant sheet which will show to the 
plant manager at a glance his various departments, and the 
total for the plant, carried at the top of his chart. Likewise, 
on a single summary chart for the entire business, the presi- 
dent or controller can see the work of the various plants, with 
a total for the entire business at the top of his chart. In 
other words, this method of charting can be carried to the 
last degree of detail or reduced to the shortest possible sum- 
mary, and the entire structure of an industry can be shown 
by a similar structure of co-ordinated charts. Once started, 
the work can be carried out almost entirely by clerks and 
secretaries, freeing the executives from routine, analytical 
work, and largely freeing any special statistical department 
for research work. The progress-chart is the “looking-forward”’ 


PROGRESS CHARTS 277 


chart par excellence, beside which all other charts are either re- 
search methods or “looking-backward” records.? 


3 The best descriptions of the Gantt charts are to be found in Mr. Wallace Clark’s 
The Gantt Chart, Ronald Press, New York City. 

The following articles in periodicals may also be consulted: 

Polakov, Walter N., Kinetic Statistics as an Aid to Production and Distribution, 
Journal of American Statistical Association, Sept., 1922, p. 359. 

Clark, Wallace, Installing Gantt Production Methods, Industrial Management, 
June, 1920. 

The books of Mr. H. L. Gantt, Organizing for Work, Industrial Leadership, and 
Work, Wages and Profit, also contain mention of the charts. 


CHapTer XXIV 
SUMMARY CHARTS 


Historical data often comes in three sets of figures showing 
income (credits), outgo (debits), and balance. A wide variety 
of names is used for these three sets, Sometimes they are 


Production Imports 


Stocks in Warehouses, etc. 


Consumption Exports 


Fig. 243. The Flow of Goods. 


called production, consumption, and stocks. Sometimes they 
are known as export, import, and foreign trade balance. 
Sometimes there are several groups of these three linked to- 
gether in a single chain of events. Thus in a single business 
concern, the purchasing agent will keep a record of his orders 
(credits), receipts (debits), and balance on order. He will also 
probably keep a record of his receipts (credits), uses (debits), 
and balance on hand of raw materials. The factory manager 
may keep a record of withdrawals from raw material stock 
(uses) (credits), production (debits), and goods in process 
(balance). He will certainly keep a record of production 
(credits), shipments (debits) and finished stock on hand. If 
goods are stocked in warehouses, the warehouse clerk will 
keep a record of receipts (shipments in) (credits), sales or 
consignments (shipments out) (debits), and stock on hand. 


278 


SUMMARY CHARTS 279 


t $ 


* 
PURGHASTNO 
orners) — AVENT sass ( Ordere 
Placed OEPARTHENT \"*Ce sve 
Poture 
emmiiments 
oe Re-" 
cetpts 
FACTORY 
Rew DRPAFTYEMT 
Matertals 
<== Pro- 
ear duetton 
Palance 
on Hand 


SHIPPING 
DEPARTMRNT 


OPERATTONS (Yisiding floz or streom atatistios = 
pertod data - eumlable) 


© 
LI 


BALANCES. (Yielding fund or etock statioticn = 
point data - not cumulabdie) 


OUTLINE 
of 
COMMERCIAL OPERATIONS 


Fig. 244, A Typical Industrial Process. 


The accounting department will keep track of sales (credits), 
payments received (debits) and accounts payable (balance 
due). Other steps may be inserted between the above. In 
these chain periods of groups of figures, the outgo (debit) from 
one deposit station will appear as the income (credit) to an- 
other. 

In fact all historical data belongs to one or another of these 
three classes. Population statistics are merely a balance be- 
tween the number of births. (income) and the number of 
deaths (outgo) in a community or country. Crop and pro- 
duction statistics generally belong to the income class, the 
figures for consumption and goods in storage or process being 
rarely available. Examples might be multiplied. 

The economists’ distinction between stocks or funds of 
goods and streams or flows of goods goes to the bottom of all 
this. For if you will regard the stocks of goods as a reservoir 
or body in repose, you will see that the income is a flow or 
stream of goods into this reservoir and the outgo is a flow or 


280 CHARTS AND GRAPHS 


stream of goods out of this reservoir. And if you are math- 
ematically inclined, you will notice that whenever any two of 
these three sets of data are given us, we can easily compute the 
third. If we know the January Ist inventory, the production 
during the year, and December 30th inventory, we can easily 
compute the shipments or sales for the year, that is, the 
withdrawal or outgo. 

Now there is a very important distinction between stocks 
and streams (whether income or outgo streams). The former, 
stocks, can only be measured at a point of time, while the 
latter, streams, can only be measured during a period of time, 
between two points of time. In the language of physicists, 
streams of goods have one more dimension than stocks, for 
they have the added dimension of time. And the result of all 
this is that while you can cumulate or total up the stream 
figures, you cannot cumulate or total up the stock figures. 
You cannot cumulate daily the population of a city in order 
to get the population monthly, for population is a stock 
figure, though you could have cumulated the number of births 
daily to get the number of births monthly. You cannot cumu- 
late your monthly balance in order to get at your annual bal- 
ance, but you must cumulate your monthly production in 
order to get at your annual production. The distinction be- 
tween stock figures and income or Onte. figures is funda- 
mental. 

The usual method of showing these three sets of figures is 
to use three curves upon a single chart. The use of three 
curves for such dissimilar data is always confusing to the 
reader of the chart, as the thin plotted line of the curve repre- 
sents in one case a static or stationary stock of goods and in 
another a series of separate and distinct additions or sub- 
tractions. Ifthe chart shows monthly data, a recasting of the 
chart on an annual basis would raise the production and con- 
sumption curves to twelve times as high a level on the chart, 
but would leave the stock or balance curve unaffected. 

The author has designed what, for the lack of a better name, 
he is accustomed to call a summary chart to show these three 
sets of figures together upon a single chart. 

This summary chart is a combination of two vertical-bar 
charts, and a curve chart in which the bars show the stream 
figures (income and outgo) and the curve shows the balance 
or stock figures. This distinction makes clear to the most 


SUMMARY CHARTS 281 


THE ACCUMULATED TRADE BALANCE OF THE 0. 8. _ 
Estimated exports, imports and accumulated (since 1600, trade balance 
United States, 1600-1920 
(Wote:- Approximations in parentheses; al) data ae of Jan. 1st) 


° o» @2 wn» & i mm e ln] o e aA og & 
Cumulated SsSsesszeegs8 SSRB8S828 FH 88885 
yalances PE ROPE AO Pa Oi he OO tC en wae oO: Si eS ee Oy Cele! Leal 6 
ee ee ce at Mie a amy DL) bee Die kg oe Rr) Oe cose Se Sees 
(- ,000,000) 1) ot x) 
Import: EOP SS NS eee a a ar oo + 2 © 2 
Leith | e2egsgtessaess388 $88 8222838238 
(-,000,000) SSS Shee ses aes Faerie 4 2-2 a 5 8 
aia a uy mm mm ££ © DOH 
fos x 
Oa SAS aie aint Ss S Wie tA es ort s 
o a NN 
Exports 2323s 332322328 885 838228325 8 3 
tad! ae eee Nar Say et el at rT at el ee a Pm)! pare) reh © ails at Cw eT 5 
Ph ee le Nas ie ee Maree" as 
~~ = ee 5] 
4 es 
30 
ea 
aa | —— hats. 
— eet 
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2¢ 
zal ‘ieee 4 a 
2 4 
3 20 
Lol 
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3 eas eae 
oe + 
e 15 (= 
8 =: 
i ae f +f i 
-_ —+ 
a 
LS 1 Ate 
10 
6 = et 


800 
05 
(0) 
16 
20/| 
26 
30 
35 
Nol 
45 
50 
65 
60. 
65 
70 
75 
80 
865 
90 
95 
1969 
05 
10 
15 
1920 


Fig. 245. The Summary-Chart. 


casual reader that the total income is made up of all the little 
income bars and the total outgo is made up of all the little 
outgo bars while the stock of goods changes according to the 
fluctuations of the curve. In practice it has been found that 
the chart succeeds admirably in showing simply and clearly 
the rather complicated relations of the three sets of figures. 
And the chart is more or less unique in its nice use of both 
bars and curves simultaneously. 

A color distinction is made between the income and outgo 
bars, income being black or green and outgo red, or income 
being white and outgo being black. A pale tint, gray or half- 
tone, is given to the entire area between the curve and the 
zero line, except where the bars cross this area. The stock 
curve or balance curve being plotted at the beginning and end 


—— 


J1EYS siyd sy 


CHARTS AND GRAPHS 


282 


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SUMMARY CHARTS | 283 


of each period of time, since these are the dates of i inventories, 
the plotting points for the curve are the ordinates of the various 
points of time. The vertical bars on the other hand are placed 
within the spaces for the periods of time (days, weeks, months 
or years) between the ordinates. Two bars appear in each 
space, the first for income and the second for outgo. These 
bars should not be any wider than is necessary to make them 
quite clear. The remaining space between bars is used for the 
plotting of the curve and for the shading of the area under 
the curve. In its finished form the curve, or stock figure, of 
balance on hand, which might be called a band curve because 
of its shaded area, forms the background of the chart. Against 
this background, the quantity added to the stock each period 
of time, and the quantity subtracted therefrom, that is, the 
bars or stream figures, of income and outgo appear in the fore- 
ground as solid bars. 

The vertical scale for the summary chart (like the horizontal 
scale) should be the same for all three sets of figures. When 
this uniform scale is used it will be easily seen that the changes 
or fluctuations of the stock curve exactly coincide in vertical 
distance with the difference between the lengths of the two 
stream bars. If the income bar is higher than the outgo bar, 
the curve of stock on hand will rise by the difference in height 
or surplus, and vice versa, if the outgo bar is the larger, the 
curve will fall by the difference or deficit. Another advantage 
of the uniform scale is that the height of the stock curve can 
be compared with the height of the various outgo bars to show 
easily the approximate period of time the stock on hand would 
suffice if income were to cease. Where the stock represents 
invested capital, a firm desires to keep this margin of stock on 
hand as low as possible, and the chart shows clearly the size 
of the inventory compared with the periodic requirements. In 
the case of non-perishable goods or semi-perishable goods, 
such as fresh food stuffs, the storage, or amounts withdrawn 
from circulation, vary a great deal seasonally, and warehousing 
conditions must be sufficient to meet the maximum stock 
which will be in storage. 

The summary chart can be made to carry a great deal of 
detail by converting the bars into compound or segmented 
bars, and the curve into a band-chart or segmented curve. 
In this case, care must be used in the shading of the segments 
of the bars, in order that they will not obscure the primary 


hey a CHARTS AND GRAPHS 


distinction between income and outgo bars. It is, however, 
possible to make all the shadings of the black or green income 
bar in various degrees of intensity by various black or green 
cross-hatched patterns, and to accomplish the same for the 
segments or layers of the red outgo bar and the green or black 
balance curve. The segments would indicate the various parts 
of the income, outgo or balance. When much detail is shown 
in this way, it becomes necessary to omit a portion of the data 


CALL 
of STOCKS RATE 
$120 1a 


100 


rs 
i\ 


1 \ iw 


i  lavVERAGE Mon, 
— CALL LOAN Sos, 
: RATE a, 
LY 
MILLIONS 
STOCK TRANSACTIONS 


SESE 
1919 1920 192] 1922 


Fig. 247. A Customary and Sound Combination of Bars and Curves. 


Average price of stocks, average call loan rate, and total transactions in stocks 
each month—Permission of Mr. Carl Snyder. 


from the chart, or to adopt such wide intervals between ordi- 
nates that the data can be entered horizontally. The latter 
method usually calls for a chart of more than usual size. 

The successful use in this chart of a distinction between 
vertical bars and curves suggests that a general rule could be 
made for all cases of complicated charts, wherever more sim- 
plicity is desired. This rule would be that bars should be 
used for streams of goods, and curves should be used for stock 
figures. The rule would have many exceptions owing to the 
convenience of the curve form in general, but it would appear 
to be a sound principle to follow, wherever the choice is open 
to us and the use of either curves or bars alone is not felt to 
give sufficient clearness. 


CHAPTER XXV 
SILHOUETTE BAR-CHARTS 


For certain purposes, only a few details of historical data 
are required for each one of a large number of different and 
heterogeneous phenomena. In price movements or stock 
quotations, for example, the practical man is interested 


GASOLINE STOCKS 
Relative Figures of Stocks on Hand of Gasoline at End of Each Month 
“As United States 
1920-1921 
(Average month, 1919 = 100) 
(Source:- U. S. Bureau of Mines) 


oawvt © @D 


Dig Ove Or aCe ips a ys rt 
OO erste OF OP 3. 20 OU i o vt MW WO 
me ctee ie et ort a 


69 
159 
145 
120 
109 

97 
105 


BONS OATTOY 


REM SRB SSN 
BEARS 


al) 
Ba 
A) 
a 
|_| 
ra 
|_| 
md 
i 
| 
a 
iy 
a 
is 
i 
a 
a 


Fig. 248, A Curve is too Detailed and Large. 
285 


286 


Course or Propuction Since 1919. 


———— eee 
RELATIVE PRODUCTION: (1919=100). 


Maxi-} Mini- 
mum| mum} 1920 
since | since | aver-| aver- 


1921 Feb.,| Mar.,| Feb.,| Mar., 


lend oflend of| age. | age. 1921.] 1921.] 1922.) 1922, 
1919. | 1919. 
oe — (| | —<—_—_ | ————_ |——— 
Foopsturfs: 

Wheat flour. 125 64 82 91 64 
Beef products... 109 67 92 83} 67 
Pork products. 151 58 93 97 | 114 
Lamb and mutton. 110 58 80 94 89 
Sugar (meltings)... 165 40 | 104 92 80 
Oleomargarine 2. 126 26 | 103 60 70 
Cottonseed oil.... 349 7{ 100) 166} 247 
mnden: 121 20 76 71 34 
Butter......... 177 64 99) 118 76 


Cheese..... --| 169 41 86 83 49 
MCC CrCAG cciccsccensceevens| 408 42) 111] 153 4 
G: 
Cotton (consumption)....| 114] 57] 109 79| 76 
Wool (consumption)......] 126 42 83 95 64 
Sole leather......,.....02- 95 63 82 719 63 
FUELS: 
Anthracite coal......... 119] 63] 101 99} 105 
Bituminous coal. 137 74) 121 89 81 


Beehive coke...... 
By-product coke... 
Crude petroleum... 


Gasoline......... . 141 98} 123} 130] 118 

Kerosene. ........ 110 71 99 

Gas and fuel oil. 136 93] 116] 127] 115 

Lubricating oil.. 135 89 | 124] 104] 103 

Electric power.... 119; 98] 113] 105] 98 
MeETALs: 


Manufactur 

LumBeEr. 
Yellow Pine... ccccccvccnccs 
Western pine............. 
North Carolina pine...... 
California white and sugar 


Douglas HT 22. «cecccesnane 
Michigan hardwood....... 
Northern hardwoods...... 
Hoemlocks\. 2.5 ccwsceceses 
Oak Hoorn? <scdeccesess 
PAPER: 
Newsprint.......secseess- 
All other paper........... 
Mechanical wood pulp.... 
Chemical wood pulp...... 
Corrugated paper board *. 
Solid fiber paper board 3.. 
STONE, CLAY, AND SAND PROD- 
ucts: 


Silica brick........ Rocce 
Clay fire brick..:......... 
Race brick<cutc-> nacccacae 
COMMENT eect kawicecainane 
Glass bottles....... AGO uC 


BUILDING EQUIPMENT: 
Baths, enamel...... 
Lavatories, enamel. ...... 
Sinks, enamel............. 
Buildings (contracted for) 

TRANSPORTATION VEHICLES: 
Automobiles, passenger. - - 
Motor trucks........-0++- 
Locomotives........+-++-+ 
SHINS s raeamecicwen tects ct 


aeeeee 


1 Since July 1, 1921. 
? As represented by tax-paid withdrawals, 
3 Relative to last 6 months of 1919. 


From Monthly Survey of Current Business. 


Fig. 249, The Essential Data, 


_ 


SILHOUETTE BAR-CHARTS 287 


primarily in the latest quotation, but would also like to know 
whether this last quotation is an increase or a decline from the 
quotations on previous dates, and how it compares with past 
maximum and minimum prices. Here, then, are four points 
of interest to him, the present, the immediate past, and the 
prior record-making peak and valley prices (regardless of the 
dates or time of the latter). And the point is that we want to 
see these facts, not for one only but for a large number of 
commodities. It would be easy enough to present curves for 
the individual commodities, or even to combine a few on one 
curve-chart, but how can we present only the facts wanted, 
for all the commodities, graphically in one simple chart? 

If you were to stand a historical curve-chart up on edge 
and view it from the side toward which the curve is moving, 
you might succeed in imagining that the curve was really 
snaking its way directly at you. And if it had actual volume, 
instead of being a thin line of ink, you would see most clearly 
its nearest end representing the last value, and behind that a 
short portion of its previous values, and still further back you 
could make out the silhouette of its extreme peak and valley 
points. Eureka! These things are all you wanted to know 
about each individual curve, and seen from the planes in 
which the curves lie, each curve compresses to the width of its 
imaginary columns or vertical bars. This suggests the method of 
graphing to which, for lack of a better one, we give the name of 
silhouetting. The silhouette curve-bar is a recent development 
in graphics and probably has not yet reached its final stage. 

Since the graph is really a projection of a large number of 
curves shooting straight out of the page toward the reader, 
something must obviously be done to lift the nearest ends of 
the curves from their other and earlier positions. The method 
of segmented bars alone gives too much flatness to a picture 
which is really a projection of three dimensions on two—a con- 
densation of a three dimension model into a two dimension 
sheet. For this reason it is obvious that the nearest ends of 
the chart stand out clearly and appear to be wider, precisely 
as if photographed from a real model. The effort here being 
to produce the effects of perspective, the portion of each bar 
which represents its latest reading or value should be of full 
width, but the earlier readings, and in particular the past 
peaks and valleys, should be considerably narrower, to give 
the effect of greater distance. 


ae es 


288 CHARTS AND GRAPHS — 


If the portion of each bar connecting the latest and the 
next previous values be kept of uniform width, it 1s necessary 
to show whether the change has been one of rise or fall, that 


May 


Jun 


July 


Dec 
Aug 


Sept 
Nov 

—— 1919 Avecege 
Oct 


GASOLINE STOCKS 
Relative Figures of Stocks on Hand of Gasoline 
at End of Each Month 
United States 
1920-1921 
(Average month, 1919 = 100) 
(Source:- U. S. Bureau of Mines) 


Fig. 250. The Same Curve Seen From Its End. 


is, which end of the bar is the latest reading. This can be 
indicated with a small arrow-head in the bar, or by solid 
shading of one color for rises and of a totally different color 
for declines, or by both methods together. Moreover the 
latest reading might be indicated by a star or other symbol 
which the reader can quickly glimpse. On the other hand, if 


“ 


289 


500 
400 


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200 


200 


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Maximum since 1919 


Minimum eince 1919 


allel 
— February 19.2 


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<—~ March 1921 
px 7 uf arc 


A Detailed Form. 


Fig. 251. 


166 CHARTS AND GRAPHS | 


we are willing to make the bar of tapering shape, then the 
narrower end could indicate the earlier, and the wider end, 
the later value. | 

A more pictorial method may, however, in some cases 
prove to be the more efficient means of flashing the story of 
the chart to the reader. Thus, if a large circle be used for the 
latest value, and a small circle for the next previous, with a 
triangle or star for the highest past peak and an inverted 


Eaad) 


KEY 


ee © average price in 1921 


yrt7 7: © Average price in 1920 


“= Averago price in other yoars EY 


eS © weximun price (since 1912) and date 


a @ Minima price (since 1912) and Jate 


100 


Round Pork Seog . > 
steak ciioos eg Butter Milk Bread Flour Rice Potatoes Sugar Goffee 


RETATL PRICES 
Relative Flyures of Animal Average Prices at Retail of Specifiad Commodities 
United States 
1917-1921 
(1913 everage = 100) 
(Source:+ United States Bureau of Labor Statistica) 


Fig. 252. Data in the Chart. 


SILHOUETTE BAR-CHARTS 291 


triangle or square for the lowest past valley, it would seem 
that the results would be easily understood at a glance. Each 
of the four values could be strung upon the same central line 
(or ordinate) serving as a connecting thread in the place of 
the bar. The two circles for recent values could be white for 
rising values and black for declining ones. Such a pictorial 
system would make possible the addition of even further 
symbols, such as still smaller circles for the second previous 
reading when this was sufficiently different from the last two 
readings, and smaller triangles for minor peaks and valleys. 
The peak and valley symbols could contain numerals repre- 
senting their years or approximate dates. The enterprising 
reader will find still other embellishments, which, so long as 
they increase the “visibility” of the facts or the speed with 
which they are flashed to the accustomed and unaccustomed 
eyes, will be justified. 

The labelling of the various bar-curves in this chart calls 
for great care, particularly when the data is heterogeneous. 
Each compressed curve should be so distinctly and clearly 


NORMAL 
100 PER CENT, 


! 
1920 : 
LOWS * MARCH 1922 


SUGA R MELTINGS 


‘WOOL CONSUMPTION [ES © © iT 
WHEAT FLOUR? iii 2 Saunas 
PETROLEUM) «- «ETE 
ANTHRACITE COAL [ae 7; TOI 
CEMENT Cee: SLE 
rin a PAS 
MEAT SLAUGHTERED aan TST - 
LUMBER Ee Se ee 
WOOD PULP (90 
BITUMINOUS COA AAT: SARIS 
PAPER eee use 210 oe GN IPRS | 
TOBACCO PP ee ie OTT 
COTTON CONSUMPTION [Saas 2 Sy 
STEEL INGOTS” [A> ieameemmemeras 

PIG IRON Psat ec 5S | 


Fig. 253. Simple Silhouette Bars Presented Horizontally. 
Production of basic commodities in March 1922, and the low point in 192] 
conipared with normal production. In cases in which March production figures 
are not available, February figures are shown.—Permtssion of Mr. Carl Snyder. 


ED U8 ev ew © Hamiis ae meee 


COMPARISON OF PRESENT WHOLESALE PRICES WITH 1920 AND PRE-WAR. 


WHEAT 

CORN 
POTATOES 
COTTON 
COTTON SEED 
WwooL, 
CATTLE, BEEF 
HOGS 

LAMBS 


WHEAT, SPRING 
WHEAT, WINTER 
OORN, NO, 2 

OATS 

BARLEY 

RYE, NO. 2 
TOBACCO. BURLEY 
COTTON, MIDDLING. 
WOOL, OHIO, UNWASHED 
CATTLE, STEERS 
HOGS, HEAVY 
SHEEP, EWES 
SHEEP, LAMBS 


FLOUR, SPRING 
FLOUR. WINTER 
SUGAR, RAW 

SUGAR, GRANULATED 
COTTONSEED OIL 
BEEF, CARCASS 

BEEF, STEER, ROUNDS 
PORK, LOINS 


COTTON YARN 
COTTON PRINT CLOTH 
COTTON SHEETING 
WORSTED YARN 
WOMEN'S DRESS GOOOS 
SUITINGS 

‘SILK, RAW 

HIDES, PACKER'S 
HIDES, CALFSKINS 
LEATHER, SOLE 
LEATHER, CHROME 
BOOTS AND SHOES 


COAL, BITUMINOUS 
COAL, ANTHRACITE 
COKE 

PETROLEUM 


PIG IRON, FOUNDRY 
PIG IRON, BESSEMER 
STEEL BILLETS 
COPPER 

LEAD 

TIN 

ZING 


LUMBER, PINE, SOUTHERN 
LUMBER, DOUGLAS FIR 
BRICK, COMMON, NEW YORK 
BRICK. COMMON, CHICAGO, 
CEMENT 

STEEL BEAMS 


RUBBER, CRUDE 
SULPHURIC ACID 


(Relative prices 1913100.) 


INDEX NUMBERS 
200 300 400 __ 800 600 700 


(BELLE FARM PRODUCTS. PRICE TO PRODUCER 
WILLE 
aes Sree S 


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LLLMLULL LLIN LA, 


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LLL 

COLL 


cael 

PLL 
ee 
ese nn 


LLL. 


LLL 

AST 
ee ETI EEE 
pu ZZ. ZEEE lM Maa Te, 
ZZ) 
VTE LL 


em tthe, 


ICLOTHING | 
pee LLL LLIN, 


PSB ZIZZIZIZIZZ ETI. Be LULU LLY TULIPS, 
PIII Lid esdiia IE 
Uiithliidliditilttl 
GLLLMLULOLULMMITLYLS A, 
CLLIZELLLLLLL 
\ LULLLELULLLLLLETL LY 
LLL 
ULI LUTTLLLLLLLULLLULL LY TLYL LLL LLD LILLY LLL LLL SELL Li 
EZ 
Lg Ld 
EZ 


MMs hag leseelised 


FUELS 
ZL 


De, sea] 
EZ Lg A 
ee eee 


GSMMMLULLMMLS EY, 

METALS 

ULLLLULLLL LULL LULU TLL 
RES 


LLL LL 
NL hg 
yea 
ea ZZ 
LLL 


BULLI 


BUILDING MATERIALS 
LLL LLUULLLLUY LILLY UPL TOUS DL Ls 
[eee a arr 


LL Lg 
ieee Zoe a) 


PEAK PRICE 
GEREN PRICE IN MARCH 1922 


WM 
ZZ. 
LIL LL 


MISCELLANEOUS 
MMMM La 


COUITLULU LLL Uhl, 


From the Survey of Current Business. 


Fig. 254, 


A Silhouette-Bar Chart Set Horizontally. 


SILHOUETTE BAR-CHARTS 293 


labelled by its descriptive title at the base of the chart. im- 
mediately underneath it, that the reader may have no difficulty 
in finding the graph for any particular item in the series, or 
in finding the title of any particular graph. Because these 
charts generally contain a very long list of items, it is well to 
divide them into smaller groups with slight margins between 
the groups, and it goes without saying that: these groups 
should be as logical and useful to the reader as possible. 

The silhouette curve-bar can be projected on either an 
amount-of-change or a rate-of-change scale. Usually the 
index numbers are used in the place of the absolute numbers, 
so as to make the items and graphs comparable, and usually 
the scales are arithmetical. But index numbers can equally 
well be shown on logarithmic scales and the latter give an 
added refinement to the chart which shows with greater 
accuracy the significance of changes, to those readers who 
understand it. 


e 


“® 1 ae oo 


CHAPTER XXVI 
INDEX NUMBERS 


It is one of the most important functions of the statis- 
tician to compare the behavior of different phenomena and 
find out whether there appears to be a relation of cause and 
effect between them. For if such a relation exists, we ordin- 
arily expect to find evidence of it in their fluctuation. If one 
phenomenon is directly or indirectly the cause of the other 
we may expect to find the fluctuations of the first paralleled 
or mirrored in the fluctuations of the second. If both are the 
effect of a third common cause, we may still expect to find a 
marked similarity between their movements. Often there is 
a delay or lag between the time of the movements of one object 
and the reaction upon the movements of the other. Thus it 
has been shown that the fluctuations in the production of pig- 
iron follow closely those in the corn crop after a lag of about 
two years. 

The science of business forecasting is largely built up on 
these comparisons. Thus if pig-iron activity follow the corn 
crop exactly after a two years lag, it is easy to see that with 
a knowledge of corn prices today we would be able to forecast 
the prices of pig-iron two years from today. Some of the 
ablest statisticians in the country are engaged upon research 
in this forecasting problem. Unfortunately the relations are 
not simple and easily established, and the available informa- 
tion is not nearly complete enough at present to make general 
business forecasting very successful. It is, however, sometimes 
possible for a capable mathematician to construct a very 
accurate forecast for an individual business or industry in 
which the determining factors are more easily ascertained and 
measured. 

When a condition of similar fluctuations (either parallel or 
mirrored) exists, the word ‘“‘correlation”’ is used for the condi- 
tion by both mathematicians and statisticians. They have a 


294 


INDEX NUMBERS 295 


complicated, mathematical process or formula for computing 
the degree of this correlation between two or more series of 
figures. The degree or extent of this correlation they indicate 
by a “correlation co-efficient.” The mathematical work in- 
volved in determining this correlation co-efficient, which 
measures the degree of correlation between two series, is long 
and complicated. There is, however, a very simple method of 
detecting the existence of correlation or similarity of fluctua- 
tion between two series, which consists of plotting the curves 
of the two series and comparing the curves by sight. Do the 
wiggles in one curve parallel or mirror the wiggles in the other? 
If the curve has been properly plotted, correlation provided it 
exists, will be apparent at a glance. By using very trans- 
lucent paper, you can subject the two curves to “light an- 
alysis,’ that is, you can lay one curve over the other, hold 
the two of them up to the light, and immediately see the 
slightest deviation of one from the other. The method does 
not give the correlation co-efficient, or exact measure of the 
_ degree of correlation, but it serves to give a sufficient idea of 
the amount of correlation for most purposes. 

The only problem is. to plot the two curves correctly. 
Ordinarily, the different series of data have different units 
of measurement. Thus corn is measured in bushels and pig- 
iron in tons, one a measure of volume and the other of weight. 
So far as prices are concerned, they read in common units of 
value, but one may lie much higher up on the scale than the 
other curve, because one may be measured in dollars and the 
other in cents. And as you know, the same fluctuation will 
be greatly magnified in a curve lying higher up on the chart, 
than in one positioned low. ‘The fluctuations might be identical 
and yet when plotted on the same scale of numbers they 
would appear very dissimilar, because of the exaggeration of 
the fluctuations in the curve of iron, lying higher upon the 
chart. 

There is however a very simple method of reducing two 
entirely different series to a common scale with a similar posi- 
tion and range onthe chart. The trick is to use “index figures.” 
In a previous chapter, the distinction between absolute and 
relative figures has been pointed out. The bushels of corn are 
absolute figures, but the percentages which these figures are 
of figures at a certain point of time are relative figures, also 
called index figures, index numbers and indices. When using 


296 CHARTS AND GRAPHS 


indices for a historical series, it is necessary always to state 
what year or time is considered as the basis for the percentages, 
that is, which year or time is taken as one hundred per cent. 
The hundred-per-cent year is called the base-year or “base”’ 
and the price or value at this time is called the “base figure’’ 
for the relative series or index figures. 

The first step in reducing a series of data to index figures, 
therefore, is to select the base. Obviously a great deal de- 
pends upon the base you choose. If you select the highest 


PER CENT 


1919 1920 i9z! 


Fig. 255. 


Sales of 57 department stores in the Second Federal Reserve District and 8 
leading chain stores doing a country-wide business (average monthly sales in 
1919=100%). Permission of Mr. Carl Snyder. 


figure in the series the rest of the series will lie below the 100% 
line on the chart; if you select the lowest figure in the series, 
the entire curve will lie above the 100% line. The common 
practice is to use an average of a number of figures during 
times which were considered normal. Thus in its “price rela- 
tives,” or index figure of prices, the Bureau of Labor Statistics 
has adopted the average price for the year 1913 as the base 
in all its price-series on the general theory that this was the 
last normal pre-war period of time. Others have adopted 
other periods of time as the bases for their series. In special 
cases you may have to adopt a special year regardless of its nor- 


INDEX NUMBERS 297 


1920 1921 


Fig. 256. 
Monthly sales of department and apparel stores in the Second Federal Reserve 
District and of mail order houses and chain stores doing a country-wide business 


1919 average = 100%) —Permission of Mr. Carl Snyder, 


298 CHARTS AND GRAPHS bie iis: 


malcy. In comparing the index figures from different sources, 
you must convert both series to a common base year. A rela- 
tive series can be changed from one base to another in the 
same way that it was constructed from the absolute series, 
that is, by dividing the series through by the value for the new 
base period. 

PRODUCTION OP MANdPACTURKD Goons 


Physical Yolums of Production and Growth of Population 
United states, 1899-1919. (Sourco:- from Mr. E. E. Day) 


All aA a + © »w © » Ue Els rely aieig baa Seat) 

mutotree § § F888 F288 RSSRSEESCEARES 
“goods — 

wTetiten, 8 G8 3338835238325 $888 g 

a - » »w A wh A OO Cee | Den fe ae ee 

ono 6 & wo na On 

wee BN a 3 8:35.93 § § Shs Gogh & g 2. ag 

CO~ ‘a. So ber a nD rt Bo = Ss » ow F © 

Foptation 3 § $ § 8 {a S 2888 8 8 338 8 3 


aD aA nN o 80 &-& © ort NN O's. te 
BSG Pek Bo bg S58 io Rae sae St ees 
ER NR Ctl AS eA: 05 eA a A ECAR OE RECT eS RES ES 


Fig. 257, Obviously, only Index Numbers are Possible. 


Many books have been written on the subject of index 
numbers.! There is nothing difficult about the task of con- 
structing relative figures for a single series of data. As already 
stated, you first select a base for the series and then divide 


1 The literature on this subject is considerable. In particular, the student should 
refer to the works of Wesley C. Mitchell; also to Irving Fisher, “Best Form of Index 
Numbers,” American Statistical Association Quarterly, March, 1921, p. 533. 


INDEX NUMBERS 299 


all the other figures of the series through by this base figure, 
turning them into percentages of the base, that is, into a rela- 
tive series. [To compare two series on a chart, you merely 
turn them both into relatives to the same base, then plot the 
curves of the two series and compare their fluctuations. The 
difficulty comes when you want to make a common index series 
for two relative series. Thus, if you have the price-series of 
various grades of steel, how will you make a single index series 
of figures for steel of all grades, that is, how will you combine 
these various relative series into one single index series. For 
if you are going to compare steel and wheat prices, it is obvi- 
ously not an easy matter to have to compare a thousand dif- 
ferent grades or kinds of steel with as many different grades of 
wheat. It is much easier if you have a single index figure for 
the price changes of all steel, and another for the price changes 
of all wheat. The problem of finding a series of figures which 


PERCENT 
s00 


250 


200 


150 


19161917 “1916 1919 ISZ0] 192i, 1922 


Fig. 258. Various Indices of the Same Phenomenon Produced by 
Different Methods of Weighting. 
Index of the prices of 20 basic commodities compared with the Department of 
Labor Index (325 commodities) —Permission of Mr, Carl Snyder. 


300 CHARTS AND GRAPHS 


will serve as index numbers for several relative series is not an 
easy one. | 
Briefly, an index for a number of relative figures must be 
some sort of an average of those figures. But there are several 
kinds of averages, each with its own particular merits and pur- 
poses. For simplicity, let us suppose that we have only two 
original series, the price of a loaf of bread in the city of Osh- 
kosh, and the price of a loaf of bread in Kalamazoo, and wish 
to find a single index for the price of a loaf of bread throughout 
the county containing these two towns (assuming that they 
together comprise the total population of one county). If 


PRICES OF OIL STOCK AND- PETROLEUM 


Relative prices of 20 oil Shares, petroleum products, 
and crude petroleum. 
(100% * 1919) 


(Source:- Pogue, Economics of Petroleum) 


and non dD hh a DH o o» o iad o ao 6 
Crude Price Sy Ghee ee rae ese alles he ig Mie ees td Py 2 5 
Reirlem cts Ty be ee ee 
Oa a a a © ©» aA a a 
Petroleum a Rl Se el Se Ponca Steet Cer mr aS 
products — a a a a a a a cl a - - a «a a | a 
vn © » @ & 2 ¢ » © 
20 Oil be fami = SEF 29 eS A SaaS hee Go oh 


shares 


INDEX NUMBERS 301 


Kalamazoo bread sells at 10c a loaf and Oshkosh bread at 15c, 
the average price would appear to be 1214c. This is the simple 
arithmetic mean of 10 and 15c. If we are using a geometric 
mean as an average, the average would be around 12c, and if 
we are using a harmonic mean as the average, the average 
would be around 13c. For most purposes, however, the arith- 
metic mean, that is, the common or garden variety of average 
will do. 

But let us suppose that Kalamazoo has a population of 900 
persons and Oshkosh only 100 persons. Hence, for every loaf 
eaten in Oshkosh there will be nine loaves eaten in Kalamazoo 
and the true average price of every ten loaves will be about 
10%c. A little study will show that the average loaf is nine 
times a Kalamazoo 10c loaf for every single time it is an 
Oshkosh 15c loaf. In other words, we must “weight” the fig- 
ures before averaging them. Now this weighting is sometimes 
a very difficult problem. How would you combine changes in 
the cost of butter and changes in the cost of other foodstuffs 


WAGES AND FAR 
Comparison ef hourly wage rates in Civil and World Ware 
United States, 1860-79 and 1913-22 
(8o0urce:~ Monthly Labor Review) 


“A @ 
ey $2828 2838 
me S88 BRK TSESERBSERSER EE SESS 
Civil: PE NP I IP a PP nap eh) A ne Nt) rN) SP) 
Wer 
gue 8 SS £2 8 S58 8.8.5 8 28S Ss sos ere 


ivil War 


60 

20 
to) 

Co > 

Wrawr SA AAAAASASREATKS SEHK SF BBB 

a 7 - -« 

o & i Say, ee ow &o fF D&D Hh 

Civil war gcoessssssss grees eF & e.8 
“ 


Fig. 260. 


CHARTS AND GRAPHS 


RO2 


“Laphug 1407 “4p fo 

UOLSSIUbdd JT— ET 6] JOF SaINSY JO sadequadoiad sv passoidxa pue[suy pue soieig powuy ay3 ur sadtid Aypowuros a[BSajOou AA 
"19% *314 
09@1 _osel 


Oc6t ~— OLE GOét 0681 OeeL O29 Ovel 


Feet OLZ6L)} ONWIONA 


G3GN3dSNS LNIWAWd JIdad$; 


| GLEL OL tIeL ; on 
ZI@t 40 UWM=SUWM DINOS 10d¥N 


to get a common index, an average change in the cost of all 


foodstuffs, which could be used as a cost-of-living index figure. 


ocet = =ozel ole OOCt O6Lb 


0 


os 


001 


393 


4919-20 


GAGES, PRICES AND BMPLOYMEND 


- 
2 
4 

191F 


| 
r| 
5 


@AGES, PRICES AND EMPLOYERS 


Fig. 262. 


1916 


100 


Reteil 
food 


(June, 1914 = 100) 
(Source:- Monthly Labor Review.) 


WAGES, PRICES AND EMPLOYMENT 


Relative figures of weekly earnings and employment in factoriae 
tn ®. ¥. Stete and of reteil food prices in the United States 


O 


*OSN 401d $0S/0T sedeg 


ew pgac addy 


304 CHARTS AND GRAPHS 


In this writing, nothing more can be done than to indicate the 
problem. It has not yet been finally answered. 

As to the plotting and other details of chart-making for 
index numbers, the rules laid down for historical charts in gen- 
eral apply. The use of index numbers, however, generally 
brings all data to a common scale and range of variation so 
that a uniform charting field should be used for these charts, 
when you are preparing a series of them. The uniformity is of 


d 
d 
4 
o 
eeeee 
Fo 
dks 


Tin Mae Nay Jy Seer, Nov. Jan. far. Clay Tuy Sarr oy. 
S919 (720 


Fig. 263. Correlation Shown by Mirroring. 


The chart shows Foreign Exchanges on New York below the heavy line (in terms 
of depreciation from parity), and commodity prices above the heavy line— 
Permission of Mr. Carl Snyder. 


INDEX NUMBERS | 305 
value for making comparisons. It is well to place the data of 
the original or absolute series beside the data of index or rela- 
tive figures in the data attached to the chart, whenever the 
relative has been computed directly from absolute data. Of 
course this is not desirable where the indices have been com- 
piled from a large number of original series. 


LIABILITY 
PRICES ‘THOUSANDS 


Fig. 264. A Slightly Lagged Correlation. 
Average liability of failures in the United States each year compared with changes 


in wholesale commodity prices. (Department of Labor index.)—Permission of 
Mr. Carl Snyder. ; 


Relative figures and index figures? can be frequently used 
for all sorts of data other than historical data, but the principles 
and applications are the same and the greatest use occurs for 
relatives and indices in historical series. They afford a simple 


2 The distinction between relative figures and index numbers 1s really very clear 
and should be adhered to. Relative numbers are directly related to absolute data; 
the absolute data is that original series of actual figures which has an individual as 
well as a collective meaning. Thus, price-quotations are absolute figures. From these 
absolute figures we derive relative figures by the process of division, using a constant 
divisor which we call the “‘base-figure.”’ But the relative figures, so derived, have no 
individual significance; they take on a meaning only collectively, as a series, each being 
a ratio between two absolute figures. Relative figures are but one step removed from 
absolute figures. ; 

Very different from these, are index numbers. These have no corresponding 
absolute data; they are merely indicators of some theoretical and wholly imaginary 
idea, such as the combined movement of many individual things. They are usually 
derived from relative figures, as explained in the text, either by simple or by weighted 
averaging. 


~ ee 


306 CHARTS AND GRAPHS 


and sound means of comparing any series, bringing widely dif- 
ferent series, or even series measured in different units, together 
into easily compared curves. They are at bottom no more 


600 


600/—— 


| | eee 
ae 


~ 4915 116 «1917 1918 =. 1919 1920 1921 1922 
Fig. 265. 


Wholesale commodity prices in four countries (average prices in 1913=100%). 
—Permission of Mr. Carl Snyder. 


than percentages, though different from the percentages used 
in 100% bars and band-charts, in that the value of 100% is 
no longer a total, but merely one of the values in the series. 
That the comparison of curves is largely incidental to the 
search for correlation between the phenomena which the 
curves represent, and that correlation studies are in historical 
series very often directed to the practical end of forecasting or 
predicting future conditions, are details which in no way limit 
the general usefulness of indices. You may be seeking light on 
the probable level of prices in your business in the future; this 
calls for forecasting and therefore for a knowledge of attendant 
and preceding developments. But you may also be only in- 
terested in the relation between changes in your advertising 


ee | aria ” Mas A 
ae case, too, - -you 


entage” or mace: 
nesuckeasIve diffe rences) of a 


ina bets ee alter See ne 
It is discussed in later chapters. 


hail 


‘a 2 


CuarTer XXVII 
FREQUENCY SERIES 


. 


We have now to consider the curves for data of a non- 
historical nature, that is, data in which time is not the inde- 
pendent variable. This is often data of conditions at a single 
moment of time—a cross-section, as it is sometimes called, of 
the phenomenon. At other times it is a compilation or recap- 
itulation of phenomena (events or conditions) through a period 
of time—still, if you please, a cross-section. The analysis of 


_ such data proceeds through a series of changing conditions, and 


the conditions can sometimes be so coherently arranged as to 
form a variable. When this is the case, the data can generally 
be profitably shown and studied by means of a curve-chart. 
Curves of this nature are called “frequency curves,” a name 
which is derived from their chief purpose, which is the display 
of the frequency with which the phenomena occur under 
given conditions. They are also sometimes called “picto- 
grams,” but the latter name has fortunately not found general 
acceptance. 

A few examples of this type of data will serve to make the 
class clear. The manager of a chain of retail stores, and to a 
CITY FINANCES 
Percapite Revenue Receipts and cost Payasnats 
United States 


Year Ended June 30, 2919 
(Source:- Tnited States Census) 


Population Pereapits Percepite 

of Cities Rerenus Sxpense 

(1917) (Dollsrs) (Dellars) 
30,000 - 50,000 27.14 25.23 
§0,000 - 100,000 26.23 27.29 
100,000 - 300,000 29.18 32.10 
300,000 - 500,000 39.53 40.18 
500,000 and Over 41.87 40.73 


Fig. 266. 


~o 
308 


FREQUENCY SERIES 309 


lesser extent any distributor over a large territory, will be 
benefited by a report showing the per capita sales in cities of 
different sizes, as such statistics will show him the comparative 
value of large and small town outlets for his goods. Here the 
classification would be according to the population of towns, 
and for certain purposes a simple analysis would show the 
average per capita sales in towns of each size. Again a manu- 
facturer is putting up his products in many different sizes and 
(48 - can) 
SIZE CASES 
1/2 pound cans —1,200,034 
1 pound cans 13,901,592 
1+1/2 pound cans 1,529 
2 pound cans 3,003 
PRODUCTION OF RED SALMON 
Output of Canned Red or Sockeye Salmon 
Alaska Fisheries 


7-year Total, 1913-1919 
(Source:+ "United States Bureau of Fisheries") 


Fig. 267.] 


cases, and an analysis of sales according to size might take 
the form of a table showing sales of each size which again 
might be charted in a frequency curve when the sizes form a 
connected mathematical series. And to take one more example, 
the manufacturer of building materials might find advan- 


RENTS IN DENMARA 
Average Yearly Rentals of Family Dwellings 
Danish Cities 
1918 and 1919 
(1 Crown at par = 26.8 cents) 
(Source:- Monthly Labor Review) 


Capitel Provinces 
1918-1919 1918 1919 
( 


room and kitchen 138 148 85 99 


r= 


rooms and kitchen 290 304 179 198 


nm 


rooms and kitchen 415 434 271 301 


a 


4 rooms and kitchen 544 566 379 420 
§ rooms and kitchen 780 828 602 457 
6 rooms and kitchen 1065 1117 632 706 
7 rooms and kitchen 1369 1464 706 85) 


8 rooms and more - 2103 2328 1033 1151 
Fig. 268. 


310 CHARTS AND GRAPHS 
tageous a curve showing the number of one-story, two-story 
and higher buildings in his territory. 

Both the historical and the frequency series are numerical 
distributions, that is, their independent variables are mathe- 
matical series or progressions. When the independent variable 
marks specific points or periods of time we call the numerical 
distribution a historical series. In all other cases we call it 
a frequency series. Nor is it possible to apply this distinction 
always, for there is a large class of numerical distributions in 
which time is counted—not from a single common origin-point 


of time but from various and usually unrecorded and unim- 
EFFECTS OF DIPHTHERIA ANTITOXIN 
Chances of Recovery due to Use of Antitoxin 


On Various Days after Diphtheria is Discovered 
(Source:- Kolle and Hetsch) 


Day of Disease Percentage 


on which of Cases in 
Antitoxin which 
is first Recovery 
Administered is Made 
4 100 
2 96 
3 86 
. VY 
8 62 
8 46 
Fig. 269. 


portant reference-points, and these also are to be classed as 
frequency series. Counts or classifications (i.e. distributions) 
of the population by age in years, or of mortality-rates, 
marriages, weights, heights, illiteracy, or the like, by ages; 
of orders or shipments by length of time taken to complete, 
or the like, are examples of frequency series which have as 
their basis, time. 

In the consideration of frequency series, we may regress a 
moment to the general subject of statistical tabulation. For 
it is in frequency series that the greatest measure of statistical 
treatment is called for, not alone in the handling of the com- 
pleted series, but in the preliminary work of compiling the 
series. And in actual practise the student will encounter a 
baffling heterogeneity of frequency distributions, presented by 
their compilers in various shapes and statistical fashions and 
often, unfortunately, in what he will come to recognize as 


FIRE LOSSES IN THE UNITED STATES 
Statistics of Losses of Property due to Fire in Larger Cities 31 I 
1918 


(Source:- National Board of Fire Underwriters) 


ager Property Lose 

©: 

State City Population Fires Total § Percapita 
nae Birmingham — 225,000 2,685 591,207 2.63 
Cal Los Angeles 700,000 3,100 1,388, 205 1.98 

Oakland 225,000 1,503 136,746 61 
Sani Premorace 825,000), S61 in reece Ween 
Col Denver 290,000 1,538 374,214 1.29 
Conn Bridgeport 200,000 752 118,499 +59 
Hartford 140,000 651 216,696 1.54 
flew Haven 175 ,000 874 266,717 1.52 
Waterbury 100 ,000 459 133,993 1.34 
O.¢c. Washington - 400 ,000 1,662 683,171 1.46 
Pla Jacksonville 110,000 A235 260,298 2.28 
Ga Atlanta 230,000 762 665 ,336 2.868 
111 Chicago 2,816,000 17,208 7,331,023 2,860 
Ind Indianapolis 300,000 2,969 1,068 ,937 3,56 
Ta Des Moines 108 ,000 925 285 ,338 2.64 
Kan Kansas City 100,000 984 156 ,765 1.66 
fy Louisville 265 ,000 919 562,204 2.08 
La New Orleans 380,000 882 548,248 1.44 
Ma Baltimore 760,000 3,244 3,205 ,602 4.27 
Mass Boston 808,310 4,934 2,577,584 3.19 
Cambridge 112,000 584 418,353 3.73 
Fall River 130,000 414 210,631 1,62 
Lawrence 106 ,000 622 78,120 74 
Lowell 118 ,000 943 232,103 1.96 
Lynn 104 ,000 631 96,099 91 
New Bedford 120,000 643 249,917 2.08 
Springfield 180,000 812 357,947 2.75 
Worcester 190,000 1,345 246,839 1.30 
Mich , Detroit 900,000 4,190 4,026,279 4.47 
Grand Rapide 145 ,000 1,093 787 , 604 6.22 
Minn Duluth 100 ,000 415 169,807 1,69 
Minneapolis 400,000 2,279 924,733 2.31 
St. Paul 276 ,000 1,299 633,140 2.30 
Mo Kansas City 320,000 3,297 1,027,052 3.22 
St. Louis 900,000 4,088 1,616,254 1,80 
Neb Omaha 205 ,000 1,233 293,445 1.43 
NJ. Camden 110,000 438 76,933 +70 
Elizabeth 110,000 410 99,013 98 
Jersey City 300,000 1,169 413,563 1.38 
Newark 450,000 1,545 896 ,881 1.99 
Patterson 130,000 473 393,197 3.02 
Trenton 108 ,000 403 287,079 2.66 
N.Y. New York. City 5,006,794 13,429 12,488,258 2.08 
Rochester 300,000 913 390,375 1,30 
Scheneotady 108 ,000 316 69,696 66 
Syracuse 160,000 567 253,527 1,58 
Yonkers 106 ,000 482 168,779 1.59 
Ohio Akron 175 ,000 835 389,819 2,23 
Cincinnati 418,022 1,468 612,742 1.47 
Cleveland 750,000 3,906 1,793 ,044 2,39 
Columbus 240,000 823 249,375 1.04 
Dayton 153,930 1,081 300,361 1.95 
Toledo 220,000 1,066 1,500,075 6.82 
Youngstown 130,000 745 199,518 1.53 
Okla Olkahoma City 110,000 510 402,080 3.65 
Ore Portland 326,000 944 552,831 1.70 
Pa Erie 112,000 425 100,267 +89 
Philadelphia 1,850,000 4,204 4,885 ,485 2.64 
Pittsburgh 600,000 2,580 1,707,007 2.84 
Reading 110,000 141 138,218 1.26 
Scranton 150,000 405 502,811 3.35 
R.I. Providence 260,000 1,766 627,611 2.41 
Tenn Memphis 166,000 1,738 650,993 3.92 
Nashville 150,000 667 406,751 2.71 
Jex Dallas 140,000 893 253,436 1.81 
Fort Worth 110,000 637 226,938 2.06 
Houston 150,000 1,003 1,010,052 6.73 
San Antonio 160,000 401 203 ,996 1.36 
Oteh Salt Lake City 130,000 572 347,066 2.67 
Va Norfolk 150,000 173 = 4,084,267 27.23 
Richmond 170,000 741 134,426 79 
Gesh = Seattle 380,000 2,358 762,757 2,01 
Spokane 157,626 728 334,617 2.13 
Tacoma 123 ,000 194 163,863 1.35 
fies Milwaukee 510,000 2,228 917,336 1.70 


Fig. 270. The Raw Material for a Frequency Series, 


Le CHARTS AND GRAPHS 


various stages of compilation. While we cannot attempt to 
cover the subject as thoroughly as it is treated in the statistical 
text-books, yet we may well give it a brief survey, in order 
that the chart-maker may be enabled the better to construct 
his frequency curve. 

The first stage in the preparation of a frequency curve is 
the simple listing or list of observations. This list is not in 
any sense a frequency series; it is merely the crude form, the 
raw material, from which the frequency series will be made. 
The stubs of the list are names, or numbers, which can be 


Check Average 
number datly 
of man output 
4003 381 
4182 380 
4206 370 
4215 392 
4220 400 
4e21 414 
4223 394 
4200 413 
42382 416 
4238 3807 
4276 892 
4282 374 
428? 847 
4289 406 
4342 377 
4850 428 
4354 390 
4356 398 
4361 a2 
4370 387 
4373 3b2 
4392 4i1 
4395 408 
4318 410 
4402 392 
4419 407 
4426 425 
"4452 399 
4465 401 


OUTPUT OF WORKERS 
(Non-stereotyped 
operation) 
(Source:- Report 
ef P. S. Florence) 


Fig. 271. Another Crude List. 


called items, and the list itself is composed of numerical values 
or other observations which have been noted for these items. 


FREQUENCY SERIES 313 


It is possible for both stubs and observations to take the form 
of numbers, but still the list does not form a series. Also it 
is possible for both stubs and observations to be abstract, 
that is, not numerical. Usually the stubs are not numerical, 
while the observations are. The length of the list, that is, the 
number of items in it, indicates the total “population” or 
“universe” of the distribution or series which will be formed. 
A universe of much less than a hundred items is not likely to 
prove a very reliable “sampling;” as a rule the sampling should 
be considerably larger and detailed reliability can generally 
be had only in samplings which contain thousands of observa- 
tions. The trustworthiness of a sampling also depends, of 
course, on the size of the external or unobserved universe, as 
well as upon bias and error in the selection of observed items 
or making of observations. 

The second stage, that is, the first step in the conversion 
of this list into a series, is the rearrangement of the list in the 
order of magnitude of observations. This is done to facilitate 


Bridgeport $0.69 Fall River 1, 62 Nashville 2.72 
Osklend 261 Duluth 1,69  Springtield,Me,z.75 
Schenectady ‘ ~66 Portland,Ore, 1.70 Pittsburg 2.86 
Caaden 270 Milwaukee 1.70 Atlanta 2.85 
Lavrence 74 St. Louts 1,80 Patterson 3.02 
Richaond 79 Dallas 1.61 Boston 3.19 
Erie 89 Dayton 1.95 Kansae City,Mo.3.21 
Lyon 81 lovel} 1,96 Scranton” 3.35 
Blisaboth 298 Los Angeles 1,98 Indianapolis 3,56 
Columbus 1.04 Newark 1,99 Oklahoma City 3.65 
Reading 1.26 Seattle » 2.01 Cambridge 3.73 
Denver 1,29 Fort Yorth 2.06 Memphis 3.92 
Worcester 1,30 Louisville 2,08 Baltimore 4.27 
Rochester 1,30 Rew Bedfora 2,08 Detroit. 4,47 
Tscoas 1.33 Rew York City 2.08 Grand Rapids 5.22 
Waterbury 1.34 Spokane 2,138 +Bouston 6.73 
San Antonio 1.36 Akron 2,28 Toledo 6.82 
Jersey City 1.38 dJeoksonville 2,28 Rorfolk 27023 
Omahs 1,42 St. Paul 2,30 
Now Orleans 1.44 Minneapolis 2.32 
Washington 2.46 Cleveland 2.39 
Cincinnatt 1.47 Providenoe 2.42 
New Haven 1.52 Chicago 2,60 
Youngstown 1.63 Birminghem 2,63 
Hartford 1.54 Dea Moines 2,64 
Kanees City 1.55 Philadelphia 2,64 
Syracuse 1.58 Trentoo 2.66 
Yonkers 1,59 Salt Lake CityZ,67 


PERCAPITA FIRE Lossés 
40 74 large American citlos 
1919 


Fig. 272. The First Step is Arrangement by Magnitude. 


a count, which will shortly take place, or to enable us to sum 
up two or more sets of observations about the same items in 
the list. Notice that we now arrange by the observations and 
not by the stubs or items. Already the emphasis has shifted 


hal CHARTS AND GRAPHS 


to what in a broad sense might be called in this crude list a 
dependent variable. The reason is that we are about to forget 
the stubs altogether and make the observations (which occupy 
the place of a dependent variable) the independent variable of 
our series. In other words we are about to distribute the data 


according to the numerical size of its parts. 
The third stage, and the step which finally yields us a 
frequency series, is to gather the observations into groups, or 


$10,000 8,400 (5) 2,220 
8, 000 3,850 (3) 2,300 (39) 
6,250 (2) 3,800 (69) 2,275 (8) 
6,000 (8) 8,260 (64) 2,270 (2) 
5,760 (2) 8,200 (85) 2,250 (22) 
5,500 (82) 8,160 (6) 2,220 
5,250 (11) 8, 100 (27) 2,200 (87) 
6,000 (82) 8,060 2,100 (52) 
4,850 (3) 38,050 2,000 (91) 
4,800 (3) 8,000 (819) 3,900 (9) 
4,750 (12) 2,800 (16) 1,800 (46) 
4,600 (2) 2,870 (3) 1,750 (20) 
4,500 (95) 4,860 (10) 1,700 (16) 
4,400 (16) 2,820 (2) 1,680 (2) 
4,380 (16) #800 (74) 1,600 (8) 
4,200 (3) 2,750 (53) 2,500 (10) 
4,150 (15) 2, 700 (180) 1, 487 
4,000 (207) 2,690 (4) 1,400 


8,937 2,650 (20) 1,250 (4) 
8,900 (3) 3,640 (8) 1,000 (2) 
3,850 (2) 2,600 (101) 600 
8,800 (7) 2,670 200 
8,781 (8) 2,520 

8,750 (49) 2,600 (197) 

8,700 (8) 2,460 (2) 

8,650 (2) 2,450 


8,600 (108) 2,480 (2) 
8,500 (117) 3, 400 (91) 
3,450 (11) 8,950 

B,487 (2) 2,340 (8) 


COLLBGB PROFESSORS’ SALARIBS 
Sslarics paid to full professors in the 
colleges end universitios (publio instie 
tutions only), United States, 1920. 
(Source:> 0,8,Burean of Education.) 

(Total number © 2,460) 


Fig. 273. A Common Tendency to Bunch Up. 


classes, and record the count of the number of observations in 
each class. Now the observations have become the stubs and 
are the independent variable, while a new dependent variable 
has been created by the counts of the observations of each 
magnitude (that is, the number of observations occurring in 
each class). Here we have clearly an arbitrary choice of the 
independent variable. Had we recorded a different feature of 
the same original items in our crude list, we should have had 
a different independent variable for our final frequency dis- 
tribution. Had we recorded two sets of observations for each 
item we should have had to make our choice between two 


FREQUENCY SERIES 316 


° 2,000 2,000 3,000 4,000 5,000 6,000 


COLLEGE PROFESSORS‘ SALARIES 
Salaries paid to Full Professors in the Collegoo and Universitivs 
(Public Institutions Only) 
United States 


1920 
(Source:- United States 5ureau of Education) 
Total Number = 2,460 
(Weter+ Scale shows amount of selery in dollars, lines show number of professors receiving sans.) 


Fig. 274. Piling Up on the Round Numbers. 


possible independent variables for the same series. When 
these alternative possibilities are presented, a wide variety of 
resulting series may be formed. The final dependent variable 
may be a count (which forms the frequency series in the strict 
sense) or may take the form of rates, ratios, percentages, or 
averages (which are only in a general sense Clld frequency 
series). It is not our purpose here to make an exhaustive 
study of these possible varieties and combinations; we present 
the reader only with a brief explanation of the simple frequency 
series (strictly so-called) in which the dependent variable is a 
count of the number of items falling within the class or group 
limits. 

It may seem at first tnought a very simple proceeding to 
gather the items into classes or groups, as above described, but 
the fact is that at this point much statistical skill is called for.! 
For the size of the groups will determine their number, and for 
the best results graphically, there should be from fifteen to 
twenty groups. But we must not only strive for a sufficient 
number of groups, but we must also consider the precise loca- 
tion of their limits. The limits of the groups affect both their 
uniformity of size and their internal distributions. ‘The last 
consideration is fully treated in the statistical authorities; in 
general, the best location of the limits from this point of view 
is one which places the largest number of the observations 


1Cf, Yule, G. Udney, An Introduction to the Theory of Statistics, pp. 79-83; King, 
Willford I., Elements of Statistical Method, pp. 105-106; also Bowley, A. L., Elements 
of Statistics, and Secrist, Horace, An Introduction to Statistical Methods. 


316 CHARTS AND GRAPHS 


COLLEGE PROFESSORS’ SALARIES 


7 Galarias of Full Professors in Colleges and Universities 
(in public institutions) 
United States 
1920 
{Soyroe:> U, S. Buresu of Education) 


(Total number of professors, 2,460. Average solary -- arithm.mean -- $ 3,126) 


$500 Clesses $400 Classes #500 Classes . K 
ass 
Totals ee Totals re Totels. Limits 
3s totals ‘i a 
Under) Under) Under Under | Under} Under Undeq mee] ae Uader 
461| 251| 362 551 | 251| 361) 452 25 
Prfe{ (> fe[ef«[ *f efefats[ [ote]: | 
y/ - - a 1 1 . 
Under 252 = We gst y 5 == 260 
61-350 1 1 i: 1 1 1 ace 
361-450 Sires 2 2 
451-560 F 1 | 2 1 1 850 
361-760 Sy eas Ne 1 Fie 760 
751-860 = a) | = | = G 2 2 
ernest : DG SY es or : oo 
[aosist,reo f=] F | 2 1, 6 $ Wose"}, a2 1,150 
Y)161-1,250 oe nal 1o | 10 n ul 1,250 
1 | w 1 -— 3 
1,261-1,360 : 1 1 a ae 22 22 1,350 
1,561-1,460 [1 ye | a2 28 1,450 
=F 22 
[1,451-1,560 | 11 |) 2) eneligs e 47 | 47 15566 
ee ca fe a BS 82 {SSS Gh 30 | . 90 1,750 
= 1 12 80 Pc 
1,751-1,850 | 46 |] cs | c6 80 | eee HF) 1,850 
f =e 146 
1,861-1,950 [9 Haq 2100 ee yez || 198 | 198 1,950 
1,981-2,050 | 91 _ll\43 [Gas 262 262 21050 
2,051-2,150 | 62 263 | 263 a sol 2,150 
2,112,260 | 110-362 [5,1 °°? || 210 zee eek 304 '|/896 cee 2280 
[2, 161-2, ee 2 506 , 
| 2.263-2,860 | 48 _baee Toa eaters BS 482 [ase | ee lees por cakes 2,360 
=2, 204 | 294 | | ar 4 | 2,450 
["23451-2,660 | 200 576 326 || 429 A190 £96 598 643 S45 2"550 L 
a 601 501 || 697 se7 |682 |_| eee 2,650 : 
oo 262 || 387 | 587 406 | 406 508 ee 2,750 
5 
2, 761-2, 860 aul jase peek i ee | SOL 601 jeeea ite Ha 726 | 2'850 
2,861-2,950 329 339 || * 459 459 = 2,950 
296123060 Seat ea 373 | 373 erect 472 ||s58 558 32980 
[3,061-3,160 | 34 |), yas | 453 453 526 | 526 sae | | 544 ee 
[sjus1-5,260 19917 [ann hes BOS Tae 223 [exe iss | = 3)250 
3,261-3,350 | 72 90 || 289 | 269 | | 306 306 ee | 3401 3'360 
$,361-3,480 | TO] 490 [ase | 207 207 Payee lPaeal siz [428 | 429 | est es 
5 sy . || 240 240 fear \S64 , 
3661234660 Ped geet jC | 208 |°? | 20s pees te eoe es 
ry $3 || 368 168 ia in 288 288 pie? 
£ | 66 66 378 378 4 
| | 221 | 22a 4) 278 \i2es | 208 Biase 
3,881-5,950 3_l 210 210 226 236 | 236 -——| — 3,950 
3,961-4,050 | 207 |i 509 | aze th a pan. || 244 244 {|__||264 264 |_| 4,050 
Tae © Ge ace “Mt as = ito eee a 287 = a 257 oe 
Ce Coe egal er Jato se casiance en jee 46 \ria us iitecee 
Farsea-a.a601 6] 3° Paap || 22 eat Oe a 132 | 182 (sand Dae 
4,461-4,650 | 95° || <> al ysaa ies Meise 128 126 125 45580 
41661-4.6650 | 2] *% Tay 109 109 il ae aig |{182 131 ates 
Gel tee Te! 34 16 ce as 20h mag tise ae aa eel 116 4/750 
781-4 ,860 5 102 
PRET calee | ase lec Nao 100 | 100 a 1980 
4,951-5,050 | 62 || > | 92 ee ez || 3 a2 || 88 oe Seca 
6,061-5,150 | = % oo Lee eH 5 
6,161-6,260 | = 1 x; | a, |__|} #2 Babies egaraines a |___|l ss fad s 
5 ,261-6,360 | 31} 4) avalos REE Pre 5 eect ees 43 43, 43 |/5;350 
6,361-5,450 |= || 32 | 32 bt ieee 43 43 || 45] 43 $5450 
6,451-5,650 | 32 || 35 s2-|| 32 se 36 | 36° 46 46 $'550 
6,561-5,660| - | 3 | 5 36 38 |! 36 35 4 38 5, 650« 
5,651-6,760 | 3 || 3 gt 3 ze 3 ih a5 5.760 
Peiteicesesot | > |— 2 3 n n|} 2 | S200 ence 
6,861-5,960 | = ]] & : 8 = 8 8 8 Beal aa) ytd ESS 
5,961-6,060 | 8 | 9 | 9 Ayla AYR 10 10 +——| 2° 10 6,050 
6,061-6,160 | - || > 2 all 20 D 
6,161-6,260 | 2] 3 |g 2 z : eel 2\| 20 eezee 
a = 2, 2 : 2 , 
6261-6 ,360 a ‘ memes | uD! 2 2 {16,350 
@,361-6,460 | - || > |. OU * . = Sn: 6,450 
| 6,461-6,660 [= | Serr . 6,550 
[seers | 
‘merece 
fies oe seee| 
es esene” | 
[8,000 | 
J eeseee Te 
fener 
|_ 20,000 1 | 
ae 


Fig. 275. Comparison of Fourteen Series Derived from the Same Data 
by the Use of Different Group Limits and Group Sizes. 


which belong to the group in or near the center of the group.? 
Thus if we are counting men of various heights, and notice a 


2 This makes each group include all observations of doubtful accuracy, such as 
the observations at and immediately about the round numbers. 


FREQUENCY SERIES 317 


COLLEGE PROFBSSORS SALARIES 
Salaries of Full Professors in Colleges and Universities 
(In Fublic Institutions) 
United States 
1920 
(Source:- United States Eureeu of Education, = 
{Note:- All dete is for $300-grours - Series G from $451 - 750, 751 = 1,050, etees 
Series H from $251 - 550, 551 - 850, etc.; - Series J from-$351 - 650, 651 - S50. ates; 
the total of each series being the same, 2,460.) 


fee 8 eS 2 8 eS 8 SS ee eg) ale 3S a 
Series ee 
: S ere BUS 
setery Sieh PRES Se 8S BGR 8S F R0Sr ili 
baad Sn ne LORE CRATE Pea eg) Na Mee Ur hia ture wes 2 
eee Sans aC CUA ONE Sa Al OA a ON ciel = ent 
Number SC SRT MN ND SONS NCL aS ° 
Series 
Lt 2 ° ° 
“w*reuny SRESEER 2229288582888 8 : 
bd GIONGTS [Cl AAT OY OT RESET BION Meee Rae ome mea leat ie & 
SS Wai RR ee SS Oe ee ee oe 
Series 8 
G ° ° SO pao ° 
be ahs Salar. $28322222232228228 88 : 
y ° ad tas So ee ee oF 8 § 


Number of Professors 


Dollars of Salery 


Fig. 276. Comparison of Curves of Three Series Derived 
from the Same Data. 


tendency of the records to bunch up heavily at the round num- 
bers (which is only natural in such measurements) we should 
do well to make our groups run from half inch to half inch, 
so that each full inch will be in the center of its group. 
Strictly speaking, the ordered list may be considered a fre- 
quency series with such minute groups that there is but Res 
stub-value (however frequent it be) in each group; that is, that 


318 CHARTS AND GRAPHS 


each group contains but one value of the independent variable. 
The series is generally unsatisfactory, however, because of its 
unwieldy length and its many omitted groups or classes (that 
is, classes with zero frequencies). We are therefore called upon 
to make larger groups that they may be fewer in number. 
As we increase the size and reduce the number of these groups, 
we find the curve becoming more smooth in outline, the zero- 
frequency groups disappearing. When carried out in detail, 
the process is very like the moving-total operation which we 
have seen performed on historical series, the same smoothing 
out of insignificant wrinkles being the result. However, unlike 
the historical series, there is no natural cyclic period to guide 
us in determining the lengths of final intervals. Hence when 
we have found the smoothest intervals, we shall take out only 
the totals (not the moving totals) for publication. If the 
results are to be published to the layman it is well to adopt 
round number intervals or class-limits, for his convenience, 
however much the data may tend to “bunch up,” as previously 
mentioned upon the round numbers. When the series is to be 
presented ,to statisticians or used in research work, and such 
“bunching up” is noticeable, care must be taken to select inter- 
vals or class-limits which will, so far as possible, place the 
round numbers near the center of each class, and the class- 
limits will therefore be fractions rather than round numbers. 

Care must also be taken that the limits of the groups be 
explicitly stated so that no confusion will result in the mind of 
the reader. ‘Thus it would be wrong to write ‘‘100—200, 200- 
500, 500-1000,” etc., in a table of the sizes of cities by popula- 
tion. Such a series should be ‘100-199, 200-499, 500-999,” 
or “101-200, 201-500, 501-999,” etc., as the case may be. 
When fractions are present, as, for example, in a similar series 
of the sizes of farms by acres, the best statement is ‘100 and 
less than 200, 200 and less than 500, 500 and less than 1000,” 
etc.; but sometimes a shorter form, such as ‘‘100-199, 200-499, 
500-999,” etc., will not be misunderstood. Whenever space 
allows and there is any doubt as to either limits or the mid- 
points of the range, two stub columns should be used, the 
first to give approximate values of mid-points of each group 
and the second to give intervals or group limits. 

Yo the feature of uniformity of size of groups or classes, 
much importance is commonly attached by statisticians, for 
the convenience which will result in plotting and other analy- 


a 


FREQUENCY SERIES 319 


oe 


sis, Obviously when a portion of the series contains groups of 
half-inch size or range and other portions contain groups of 
whole inch size or range, the two kinds of groups are not di- 


PERCAPITA FIRE LOSSES 
in 74 lerge American cities 
1919 


Percapita Number 
Fire of 
Loss cities 


Under $,60 
$0. 61-0. 75 
0,.76-1,00 
1,01-1, 25 
1, 2641, 50 
1,511,756 
1, 76-2,00 
2,.01-2, 26 
<. 26-2, 60 
¢.61-2, 76 
2.76+3,00 
8,01-3,25 
3. 26-3, 60 
3.61-3, 75 
3.764, 00 
4.0124, 25 
6. 26-4,50 
4,51-4, 76 
4. 76=5,00 
5.015. 25 
5. 265.50 
§.61-5.76 
6. 76-6,00 
6.01-6,26 
6. 26-6.50 U 
6.61-6. 75, 
6.76-7,00 
Over 7.00 


Fig. 277. The Frequency Series. 


t 
s 


me 


Ct a ee ee Oe a oS le Oe Cl a “a ou) 


rectly commensurable and comparable. It is therefore always 
a relief to discover that the compiler of a frequency series has 
been able to adopt groups or classes with regular intervals 
between their limits. The fact remains, however, that with a 
very large proportion of business and sociological data the uni- 
form group distribution is neither convenient nor satisfactory. 
There are cases in which the entire range, as it is called, of the 
distribution or series, is very great, and the great mass of 
observations occur near one end. To show the nature of the 


3“The general rule that intervals should be equal must not be held to bar the 
analysis by smaller equal intervals of some portion of the range over which the fre- 
quency curve varies very rapidly.” —Yule, G. Udney, dn Introduction to the Theory 


of Statistics, p. 83. 


320 CHARTS AND GRAPHS 


distribution through this densely “populated” portion of the 
range, small groups or intervals must be adopted; but to pre- 
vent an excessively long and tedious, often fruitless detail in 
the remainder of the series, larger intervals must be used in 
the sparse portions of the distribution. In such cases we are 
obliged to alter the sizes of the intervals, Sometimes the 


DURATION OF STRIKES 
United States 
1921 
(Source:- Monthly Labor Review) 


Days of Duration Eee 

° 
Approximate Range Strikes 
1/4 O- 1/2 32 
24 1/2 - 1-1/2 25 
2 1-1/2 - 2-1/2 42 
3 2-1/2 - 3-1/2 43 
4 3-1/2 - 4-1/2 43 
i] 4-1/2 - 5-1/2 32 
6 5-1/2 - 6-1/2 32 
7 6-1/2 - 7-1/2 41 
8 7-1/2 - 8-1/2 27 
9 8-1/2 - 9-1/2 18 
10 9-1/2 - 10-1/2 40 
ll 10-1/2 - 11-1/2 18 
12 11-1/2 - 12-1/2 11 
13 12-1/2 - 13-1/2 14 
4 13-1/2 - 14-1/2 24 
15 - 18 14-1/2 - 18-1/2 69 
19 - 21 18-1/2 - 21-1/2 42 
22 - 24 21-1/2 - 24-1/2 16 
25 - 28 24-1/2 - 28-1/2 30 
29 = 31 28-1/2 - 31-1/2 31 
32-35 31-1/2 - 35-1/2 34 
36 = 42 35-1/2 - 42-1/2 50 
43 - 49 42-1/2 - 40-1/2 37 
50 - 63 49-1/2 = 63-1/2 17 
64 - 77 63-1/2 - 77-1/2 57 
78 - 92 77-1/2 - 91-1/2 55 
92-199 91-1/2 - 199-1/2 165 
Over 200 199-1/2 and over 42 
Total 1,147 

Fig. 278. 


entire range is so excessively great that no two groups can be 
of the same size, and it is necessary that the groups increase 
progressively throughout the series. 

In dealing with such unevenly-grouped series, the analogy 
of the historical series is useful. What would you.do if asked 
to make a curve of a historical series, let us say, the world’s 
production of gold since the voyage of Columbus, in which the 
data covers at first centuries, then ages, then decades, then 
quinquennial periods, and lastly, individual years. Clearly you 


Me Sag 
££ ? 


” 
poir 


Fig. 279. 


= might sum up the parts of centuries into totals or even movirg 
} totals for hundreds of years and so zet a curve of 100-year pro- 
duction. Or you could divide the earlier data so as to get the 


CD Ooclice OF TEE ED 
Eetinzic’ 


W432-itis 
Geowce~ Teited Grete Heticticel seetrect) 
e 


1901 - 1605 1,612,098,600 
1908 - 1919 2, 167,504,500 
4911 - 1915 2,295,259,678 
1916 M4, VIE,SOO 
1917 419,422,100 
1912 222,505,552 
ie 265,166,077 
Fig. 250. 


average 10-year production in the earlier periods and sum up 
the latter portions into ten-year groups, so getting a curve o¢ 
production by decades. The intervals can be chosen at what 


¥ we «oa 


Rpp) CHARTS AND GRAPHS 


Luo'get'gos tl LL0' 991‘ G9¢ 6tet 
zag‘so9‘cee = T 239'909' eee cick 
Oot‘2zp'6tp =T OOl22b' 61h LT6T 
Oos'oLT‘ 9b «6 00S‘ SLT‘ ¥sb 9T6t 
oe6'TLt‘6sh 9 929'699'S62°2 $T6T-TT6T 
o96‘o2s‘esh 9 008'H09'L9T'2 O16T-906T 
Oz‘sts'zze O09'R60'ST9‘T  sO6T-LOBT 


O60'H2zt'OT2 OT  006'O8Z‘TOT'2  006T-T68T 
090'96%'LOT OT 009‘0S6‘¥L0'T  OB9T-TBAT 


op6'99z'90T OT 009'699‘L90'T  O88T-TLaT 


009'TOS'92T ot O00‘ gTO‘E9z°T  OLET-T98T 


O99'sbe'hs 02  000'606*969'T  O99T-THET 


Q00'999'TT 02 G00" 02e*822 OvsT-12Ze1 


OgL' OTL" 02 = C00 Tz vET OZST-1O8T 


090'929'2t OT += 00" #09‘z92"T —oOgT-TOLT 


Yeors 


Fig. 281. 


1493-1919 
tited Stetes Stetistice] Abstract) 


GOLD PRODUCTION OF THE WORLD 
Estimated 


(Source: 


O9t's90'o OOT O00'STS* 909 QOLT-TO9T 


O60'96L"» GOT 000'089' Tos OOST-cEdT 


e 8 
3° - 
3 - no 
6 Cie 
mi ee tek 
r ~ 
geo r 8 Cal 
Sea Ds che Cas 
Eo a ek ts 2 
SEs ge o 8 ri 
an a ee i 


evet size you wish, the point is that you must, for the sake of 
the curve itself, convert the data into equivalent data for uni- 
form intervals of time. 


FREQUENCY SERIES 323 


19901) 998" 84'S 


GTR GM F010 ¥ CC0T)—ces"t9 


@ oe (86€ -009) 219" 6al 


eT be 54 (669-092) 269'9L1 | 


g 
Number of Acres 
Fig. 282. 


SIZB OF FARMS 
in United States, 1920 
(Source United States Census) 


oO32"9 98 (892-921) 961° OCS Ei 


zee’Ot «94 (911-001) 699649" 
oor 
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Precisely the same operation must be performed on the 
frequency series with irregularly-sized groups. The fact that for 
curves of cumulations (either of frequency or historical data) 
this regularity of intervals is not necessary, sometimes makes 
the cumulative curve, which we shall discuss in another chap- 


1921 


DURATION OF STRIKES 
United States 


t 
~—_ 
> 
so 
[7 
- 
° 
3 
2 
5 
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= 
. 
p 
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Aworage Frequency 


in Sub-groups 


€3°5 


A 
et 


e 


NA FMF HAR RHA AHH HH EOIN SE 


N 
- 


Number of Equal (1-day) 
Sub-groups in Groups 


GOl GPT 2/T-66T ° 2/T-t6 


$9 


Lg 


LL 


Totel Frequency in Groups 


9°16 © 
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2/1-29 - 2/T-6% 


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2/t-2h 
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2/T-Te 
2/T-82 
2/T-¥% 
2/T-T2 
2/T-8T 
2/T3T 
2/T-8T 
2/T-2t 
2/To1t - 
2/TOl 
2/16 : 
2/T-8 = 2/t-n | 4 
2/T-L = 2/T-9 | 
2/T-9 = 2/t-s 
2/1-S - 2/tey 
2/1-b = 2/1-¢ 
2@/T-e = 2/t-2 
@/-z = 2/1 
U/T-T - 2/1 

27/t -0 


Groupe (Length of Strikes) 


FREQUENCY SERIES 325 


ter, far more convenient; but if we are to plot the uncumulated 
frequency curve, and have unequal classes or groups we must 
calculate the equivalents for equal intervals before we can 
plot the curve. And if, as often happens, a terminal class 
(group at one end of the series) be indeterminate, that is, have 
no maximum limit, so that we do not know its group range 
and cannot compute an equivalent figure, then we must simply 
omit the last group from the series, leaving the reader of the 
chart to conjecture that the curve runs off towards infinity. 


CuarTrerR XXVIII 
FREQUENCY CURVES 


Statisticians make a distinction between “discrete” and 
“continuous”’ frequency series, which to the chart-maker is of 
some assistance in determining the plotting points of the data 
in a frequency curve.! Discrete data is that in which the inde- 
pendent variable proceeds by leaps and bounds, lighting usu- 
ally only upon the whole numbers or integers. When this 
last is the case, the series may be said to comprise “integral 
variates.” Thus buildings may be classified by the number of 
stories or rooms they contain. Here you will find only regular 
intervals of one integer each, since fractional stories (barring 
the so-called half story) and fractional rooms can hardly be 
said to exist. Leaves may be classified by the number of their 
ribs, flowers by the number of their petals, sales-forces, de- 
partments, and establishments by the number of their em- 
ployees, and cities by their populations. All of these cases are 
examples of discrete series. 

Continuous series are those in which the phenomena may 
vary by infinitely small gradations, the data comprising what 
are called “‘graduated variates.” Thus, of we examine the 
height or weight of human beings, we find them varying by 
the smallest possible amounts. Property classified as to value, 
crops as to volume in bushels, tons, or the like, farms as to area 
in acres, square miles, etc., and sales as to sizes, are a few ex- 
amples of continuous series. In business and economics much 
the greater part of frequency data is of this type. And while 
it is not always so, yet it is ordinarily in business statistics true 
that continuous series require irregular group-ranges and 
intervals, discrete series usually falling into equal groups. The 
Nieemetion between discrete and continuous data becomes less 
sharp in those cases of discrete data which cover large 


1Cf, King, Willford I., Elements of Statistical Method, p. 106. 
326 


FREQUENCY CURVES 327 


ranges, such as cities classified by populations, for here it be- 
comes necessary to adopt arbitrary groupings which are sim- 
ilar to continuous data groupings. For small ranges the dis- 
crete data usually requires no arbitrary grouping together, as 
it automatically groups itself, and the continuous data is dis- 
tinguished by the fact that group limits have to be arbitrarily 
set for its distribution. 


MEMBERSHIP OF STRIKES 
Number of Persona Involved in Strikes 
United States 
1921 
(Source:* Monthly Labor Review) 


Striking Number Group- Average 
Persons of Range No. in 
Involved Strikes in Unite of Equivalent 
10 Persons 10-Person 
Class 
x y Dx/10 y/(Dx/10) 
1-10 219 1 219 
11 - 25 286 1.5 190.6 
26 - 50 252 2.5 100.8 
§1 - 100 214 5 42.8 
101 - 250 216 16 14.4 
261 - 600 153 25 6.12 
601 - 1,000 101 50 2.02 
1,001 - 10,000 126 900 0.14 
Over 10,000 14 ane AoE 


Fig. 284. Period Data. 


For the cnart-maker, the more valid distinction of fre- 
quency series is between what might be called “point-data” and 
“period data.”’ The former, point-data, is that which refers 
to isolated, separate, and non-contiguous points along the range 
of the independent variable. The mortality rates at various 
ages are of this type and to this type belong a large class of 
continuous series which comprise rates, ratios, percentages, 
averages, and other comparisons between different basic fre- 
quency series. Most discrete series may be classed as point-data. 
Period data, to which most continuous series belong, is that 
which covers connected and conterminuous groups or classes 
along the range, applying throughout the groups from one 
group-limit or interval to the next. The student is already 
familiar with this distinction in the matter of historical series, 


328 CHARTS AND GRAPHS 


in which flow or stream figures, for example, cover periods of 
time and stock or fund figures, for example, refer to points of 


SIZE OF FACTORIES 
Manufacturing Establishments 
Classified as to Numbersof Employees 
United States 
1914 
(Source:+ United States Census) 


Number of Establishments Employees 
Employees 


per A 
Establishment Number Percent Number Percent 


° 32,856 HI.9)  susaSereaie’ sates 
1-5 140,971 61.1 317,216 4.5 
6 + 20 64,379 19.7 606.594 8.6" 
21 - 50 22,932 8.3 742,529 10.6 
61 - 100 11,079 4.0 791,726 11.3 
101 - 250 8,470 3.1 1,321,077 18.8 
261 - 500 3,108 1.1 1,075,108 15.3 
501 - 1000 1,348 5 926,828 13.2 
Over 1,000 648 +2 1,256,269 17.8 

Total 275,791 100.0 7,036,337 100.0 


Fig. 285. Period Data. 


time. Needless to say, point-data is normally plotted upon the 
ordinates of the chart; period data is normally plotted in the 
spaces between the ordinates. 


VALUES OF MANUFACTURED PRODUCTS 
Establishments Classified as to Value of Products, with Number of Employees in Same 
United States 
1914 
(Source:- United States Ceneus) 


Establishments Employees Walue of Products 
Number Peroent Number Percent Dollars Percent 
Less than $6,000 97,061 36.2 129,623 1.8 233,381,081 1,0 
$5,000 and less than $20,000 87,931 31.9 429,037 6.1 905 ,693,168 3.7 
$20,000 and less than $100,000 56,814 20,6 999,600 14.2 2,650,229,411 10.6 


$100,000 and less than $1,000,000 30,166 10.9 3,002,071 42.7  8,763,070,135 36.1 

$1,000,000 and over 3,619 1.4 2,476,006 35.2 11,794,080,929 48.6 

Total 275,791 100.0 7,036,337 100.0 24,246,434,724 100.0 
Fig. 286. Period Data. 


We come now to the last, and what is ostensibly the most 
important division of frequency curves, namely whether the 


FREQUENCY CURVES 329 


curve shall be in staircase form or smoothed. The staircase 
form, which really represents a collection of vertical bars, is 
often called a histogram, and the smoothed-curve a frequency 


KOURS OF LABOR 
Number of Wage Earners Employed in Manufactures 
According to Prevailing Hours of Labor 
United States 
1914 
(Source:- Statistical Abstract) 


Hours of Employees 
Labor : 

Number Percent 
48 and under 831,779 11.8 
Between 48 and 54 944 ,562 13.4 
54 1,813,079 25.8 
Between 54 and 60 1,547,374 22.0 
60 1,484,662 21.1 
Between 60 and 72 249,026 3.5 
72 106 ,080 1.5 
Over 72 59,775 a) 


Fig. 287. Point-and-period Data. 


polygon. Of course, as we noticed in historical curves, the 

staircase form, if the number of steps or bars be great enough, 

will closely approximate the smoothed curve in appearance, 
ECONOMICAL SPEEDS OF TRUCKS 


Maximum Speeds at which LoadedeTrucks can be Driven 
Without Reducing Life of Tires 


Truck Miles per 
Tonnage Hour 
3 and 1 19 
‘ 1 

= 17 
a 2 
2 15 

Be eye 13 

© 2 : 2 
4 11 
5-7? 9 


Fig. 288. Point-and-period Data. 


and there remains little reason to maintain it. But for series 
of but a few items or number of groups, the distinction between 
the two is great and each has its special advantages and proper 
uses. 

The staircase curve or histogram is always more accurate 
for period data, in that it preserves the exact areas underneath 


B30 CHARTS AND GRAPHS 
the curve between each set of ordinates or group-limits. The 
reader who recalls the belittling of area-representations for 
charts in which we have early indulged in this book, may find 


PERCAPITA PIRE LOSSES 
In 74 Large Amorican Cities 
1919 


4 
© 
ed 
? 
“ 
5) 
410 
3 
5 
a 
: aan | 
0 ms ea 
ais g g g : 
: 8 rs 8 ° 8 ° 8 Gi ir 2 
- - Nn nn » o ?- = 0 o o 
Dollars of Percapita Loss 
Fig. 289. 


Showing how the smoothed curve varies from the staircased curve. The added 
triangles (dotted) are equal to the deducted ones (black) in the aggregate, but 
are not equal between any two adjacent ordinates. 


this feature to be of little consequence. But statistical prac- 
tise has it that the area is important in frequency curves. Of 
course, if we plan to apply a planimeter or other area-meas- 
uring instrument to the chart, the area is of real importance. 
Otherwise it is generally to be relegated to the limbo of aca- 
demic and scientific interests. We should, however, bear it in 
mind, that we may the more correctly interpret our charts and 
base analysis upon them. 

Not only is all period data more accurately represented by 
the individual group-areas under the staircase-curve than by 
the individual group-areas under a smoothed curve, but also 
much point-data, if we class discrete series as point-data. To 
be strictly accurate, discrete data should not be shown by a 
connected curve at all, but by separate bars; for there are no 
intermediate observations and the connection-line which forms 
the curve has no meaning over intermediate spaces on the chart. 


FREQUENCY CURVES ee ce 


Number of Women 


Number of Women 


Children per Woman ANMTDORF ORS 7 
Ge a Children per Woman 


SIZE OF FAMILIES 
Number of Children of 1,000 Women 
(married at least 15 years and having at least one child each) 
British Peerage Statistics 
(Source:- Yule, Theory of Statistics) 


Fig. 290. The Staircased Form is Appropriate. 


The chart-maker has largely to use his own judgment for plot- 
ting discrete data, as he can almost equally well, for different 
purposes, use the different methods of separate vertical bars, 
staircase or bar-like curves, and smoothed curves, not to men- 
tion plotting upon or between the ordinates. 


By C. B. Davenport, Permission of Popular Science Monthly. 
Fig. 291. A Very-Simplest Staircase Curve. 
Showing the distribution of scallop-shells by number of ridges. 


The disadvantages of the staircase form are many. In the 
first place, as in historical curves, it is more difficult to distin- 
euish a number of curves brought together for comparison 
when they cross each other frequently. In the second place, 
for period data, though not for discrete data, it is less significant 
than the smoothed form. For while the data changes abruptly 


332 CHARTS AND GRAPHS 


from group to group, the phenomenon observed usually changes 
gradually, the values usually merging between groups. This” 
is the more obvious if by a rearrangement of the original data 
we produce more and smaller groups, for then the new groups 
created take intermediate values. So to chart this data by a 
staircase curve is to give a wholly meaningless sudden change 
between groups, while to chart it by a smoothed curve is to 
bring out to the readers of the chart more clearly the gradual 
nature of these changes. In short the smoothed curve or fre- 
quency polygon has a truerx significance than the staircase form 
or histogram, for period data. 


EFFECT OF TUBERCULOSIS UPON LENGTH OF LIFE : 
Expectancy of Life in Years for White Males with and without Tuberculosis 
and Consequent Shortening of Life Due to the Presence of Tuberculosis 
Metropolitan Life Insurance Compariy Industrial Policy-holders 
1911-16 
(Source:- L. I. Dublin, Costs of Tuberculosis) 


Loss due to » © @®@ © F&F 2 DO KN & ono RR eA ° . - 
° ° ° . 5 r4 : ° : ° . ° . : 
Tuberculosis om mn Mm nm NHN A A 
Without BALD O CES AS ZO at Gio Qo VeRO Vea + Ores els 
A . 
= a - Oo td a o nn ao vr cal o + nN e » Nn nN 
Tuberou By DEAN OSD We ANG. OR AOD) COR dell a washes oe o 2 ae 


losis 

With 
Tubercu- 

losis 


Expectancy of Life 


Age in Years 
Fig. 292. The Smoothed Form is Necessary. 


For continuous point-data, that is, for point-data other 
than discrete series, the smoothed curve is often the only pos- 
sible form, the staircase form being out of the question. For 


— ; 
FREQUENCY CURVES 333 


as the data represents observations at isolated points only, 
along the range (the x-axis scale) it is to be assumed that for 
intervening points intermediate values obtain, and the stair- 
case form would be not only lacking in significance, but also in 
accuracy. The dependent variable in continuous point-data 
is usually the resultant of a process of comparison of two or 
more different frequency series, being ordinarily expressed as 
a rate, percentage, average or other ratio—a series of fractions, 
if you will, in which the denominators are not constant. Point 
data is to be found in a wide variety of forms, but is almost 
always in essence a derived series of this sort. The processes 
which yield point-data cannot be described as simply as the 

BANK SALARIES 

Salaries of Federal Reserve Bank Employees under $5000 
New York City 


1919 
(Source:- Federal Reserve Bulletin) 


as 
ndividual 1 ee wm A OF tea 
ey — 


20 
18 
(sz) 


Heads of EE Or oie une 
Families = 
(1901) 


600 


400 


Number of Enployees 


200 


Lie] Hil 
os 


S4S 2818/88 4ecees 
Dil Gh 00 1 > et eS 1s 
na A wt | « 


m 
nn > -” 


be 
° 


Dollere of Salary 
Fig. 293. It is Difficult to Compare Two Staircased Curves. 


processes which yield period data; for they are also of almost 
unlimited variety and we shall not attempt their discussion. 
It is worthy of notice, however, that during such preliminary 


334 CHARTS AND GRAPHS 


steps in the comparison of two frequency series we commonly 
find the gun-shot plotting method useful. 


(ot) (2)  4/(2049) 
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Rumen cues 
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The use of the term frequency series for point data, re- 
sulting from the comparison of two series, is permissible only 


A Cumulable Series. 


Fig. 294. 


FREQUENCY CURVES 


335 


in a broad sense, the point-data series not being a distribution 
displaying the frequencies of the phenomena in the specified 
groupings. Such point data, like the balances, stocks-on-hand, 
or fund figures, in historical series cannot be cumulated, and 


29°sh 
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Te°ay 
ure 
stay 
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,024 White Children from Birth to 6 Yeers of Age 
‘e@ Bureau) 
e a 
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eroo 
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(Source:+ Children 
+ 
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o 
n 


QTATURE AND WEIGHT OF CHILDREN 


Average Reight and Weight of 167 
+ 
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aa ten 


the connection of the plotted points by a curve is a symbol of 
the changes of the same phenomena through different condi- 
tions, not a short-hand method of indicating different and dis- 


tinct quantities. 


OTs §9e°Os q 
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Oannmenorae 


Months 


Second 


Pivot 


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A Non-cumulable Series. 


Fig. 295. 


336 CHARTS AND GRAPHS 


The question of staircase and smoothed curve plotting 
methods have been given a somewhat lengthy treatment be- 
cause it has afforded an opportunity to consider the dis- 
tinctions between discrete and continuous series and period 
and point data. As a matter of fact if there be but a sufficient 
number of intervals or groupings in our series, the distinction 
between staircase and smoothed curve plotting disappears, 
and both forms of charts become alike and merge into a third, 


PERCAPITA FIRE LOSSES 
In 74 Large American Cities 


Number of Cities 


° jo] 

So nee 

° ° ° 
+ ire} 
s 


i¢) 

50 
00 
1.50 
2.00 
2.50 
00 
50 

. 

Over 


be) Le 
Dollars of Percapita Los 


Fig. 296. A Rounded Curve. 


the rounded curve. The rounded curve is the true frequency 
curve and is superior in significance even to the smoothed curve 
or frequency polygon as the latter is to the staircase (rectilinear 
or bar-form) curve or histogram. For the rounded curve not 
only gives gradual change of values between plotted points, 
but it also gives gradual changes of the rates of change of these 
values. The derivative of a frequency polygon would be a 
histogram, but of a rounded curve would be another rounded 
curve or at least a smoothed one. 

We have not, however, laid much emphasis upon the 
rounded curve, because if the data be sufficiently detailed it 
will be approximated by either staircase or smoothed curve 
plotting. Some authorities recommend the artificial rounding 


FREQUENCY CURVES 687 


WORKMSN'S COMPENSATION 
Delays in Making Payment for Claims in Threo States 
New York (stats and private insurance) 
Pennsylvania (state and private, except self insurance) 
Massachusetts (private insurance only) 
(Source:- Monthly Labor Review) 
(Figures show Fercentage of total cases, 137 in N.Y., 4,093 in Pa, and 186 in Mass.) 


=~ =~ = 
wor rene Dawonnnanny wo wo en oy ao "4 
Mass Le ROS OC Oe Sate to Den a et fe oO . en a 
ON eFC HM OT DANN OCO ut oo we 
ANN ~ ° ~ 
~- 
= 
Ar DOH DN MANNOAHDA nw ° 
Pa se eile marion iy +O ° 
CO ArT NUN NNHRH OK YF GF | n+ N 
Cs I oe oo I ce | riw 
nan 
= ~ wo 
NHwW FH OHO Dw oD ete} ad ~ 
8. ¥ OCF eRe (ELa), Ce aer ew . i 8 ee fo) or 18) . 
Nn ODDO FF HAt D © aw PaaS: wore ” 
a wt a — 


Percentage of Cases 


Over 


Number of Days 
Firet Three Months Fourth Fifth Sixth 


Fig. 297. 
Computed averages must be used for the irregular intervals. 


of curves.? This is to be done either free hand, or by a curving 
ruler (called by draughtsmen a “French curve’’), taking care 
in either case to,pass the curve through all known values (i.e. 
plotted points), but making the remainder of the curve as 
little angular as possible. The result is almost always more 
interesting to the casual reader, obviously because of the more 
faithful portrayal of the nature of changes from interval to 


2“The object of smoothing is to eliminate accidental variations and establish 
normal tendencies.” —King, Elements of Statistical Method, p. 108. 


338 - CHARTS AND GRAPHS 


AGES OF HUSBANDS AND WIVES 
Probable Age of Wife according to Age of Rusband 
Greet Britain 
1901 
(8ource:- Yule, Theory of Statistics) 


29.1 
32.5 
38.8 
43.6 
48.6 
63.3 
68,2 
62.9 
7 
11.9 
76.6 
79.5 
80,0 


ie ek 
Upper Quartile a # 


6 

31.2 
37,3 
40.6 
45.6 

60.2 

54.9 

59.4 

63.7 
67.7 
N21 
76.0 
76.0 


2 8 
Median a 2 


90 


70 


= 


Age of Rifo in Yeers 


Age of Husband in Years 


Fig. 298. A Zoned Frequency Curve. 


interval. But it is a dangerous practise for the beginner to 
round his curves artificially, being a wholly inspired embellish- 
ment and often leading to slipshod execution. In the early 
work of the student, the smoothed curve is all that should be 
attempted, for it is all that the data establishes. And in the 
research oflice, it is for the same reason about all that is neces- 
sary or safe. In fact, in the research office, it is often sufficient 
to plot the points only and omit altogether their connections 
which form the curve, since it is often desirable to superimpose 
thereon rounded curves of a theoretical nature.% 

In general, the problems in the graphic presentation of 
curves arise, first, in the selection of the independent variable 


’ Needless to say, composite curves may be drawn for frequency series as for 
historical series, in the various forms which have been described (see Chapter XIX). 
Thus, we may have relative and absolute band-charts, gun-shot plotting, and even 
vertical and horizontal bar-charts. 


FREQUENCY CURVES 339 


FEMALE ACCIDENT MORTALITY RATES 
Death-rates per 100,000 of Population of Females of Eaoh Age 
For Specified Accidente 
United States 
1910-1912 
(Source:- Mortality Statistics, United States Census) 


. iy ~ o © © ” ve rm ~ o o ~ 2 o 
= ; ° : ° ° 2 : ° . > rs ° 
412 Accidents SC NI Li AO On. ah ear! eae Les ° 
Se at HN Kes tM) Tet ewe) les bee 88 © 
Nn 
CE Oe OM en en i a) se 2 Ss * 2 
- : ° : . . ° ° - > : 

Miscellaneous Bel ED ORS a) Brae a ati Nag eS ON Mea RU Br Tie ect Sort gr & 
o Nx CoN a RCA ea ire) my DTS We eg) 2 
. a coo) rn nw ~~ Se Oe St oF a Les, Cod 2. 
Railways DEP DS) See Tenet! ek, ic ea eg ests ap © 
( o ono wo ¢ Nn a RR OTe? vd & 
Drowning SoM CEN UNC Sarid fet) lad cee aestt Weegee SS a fo) 
~ n n_ Ss. on ow fo} A EO et eo ° ~ 

. . : . : : ° ° 4 
Palje SRR SRS eae acty ra RN) ana bes x z 
4 


Over | 


Years of Age 


Fig. 299. A Frequency Band-chart. 


and the processes of compiling the frequency Series ; second in 
the establishment of group limits for the groups in the series 
and the conversion of irregular intervals into corresponding 
equivalents; third, in the plotting of data on the ordinates or 
‘between them and lastly, in the use of the staircase curve (his- 
togram) or smoothed curve (frequency polygon). To the solu- 
tion of these problems the distinctions between discrete and 
continuous data (the integral and graduated variates) and be- 
tween period and point data bring some assistance, but no set 
rules of thumb can be given, to which exceptions may not be 
found. In the wide field of frequency curves, to which the 
historical curve stands in the relation of a small but important 
part, the chart-maker and the statistician must rely eo 
upon native judgment and the precepts of his individual ex- 


perience. 


349 CHARTS AND GRAPHS 


OUTPUT OF WORKERS 
Reloetive figures of averege daily output of manunl workers 
in stereotyped and non-stereotyped operetions 
(Totvl number of workers; stereoptyed, 21; not steroot.29) 
(Source:- P. Sargeit Flcrence) 
(Output cf median worker = 100 ) 


Bterectypsd BORE De ME OT PE A IE ONE TD Cee haa we ew 


Number of workers (percentage) 


Volume of output (relative figures) 


Fig. 300. 


A “Relative” Frequency Curve is as possible as a relative historical one. 


CHAPTER XXIX 
OGIVES 


If moving totals are your strongest weapons in the analysis 
of historical statements, cumulation is your trump card in the 
analysis of a frequency series. Where the series 1s continuous 
(as explained in the previous chapter) the process of cumulation 
does away with the need for the staircase curve, and gives us 
a smoothed curve which is far more convenient. And for both 
discrete and continuous series, the curve of the cumulated data 
is one which can be easily compared with similar curves, re- 
gardless of differences of scale figures or group units. The 
curve of cumulated frequency is called an “‘ogive,” from its 
resemblance to the outline of a shoulder. It runs diagonally 
across the chart, generally in an “S’’-shape. 


SIZE OF FARMS 
Dnited States 
1920 
(Source:- Census) 


Simple series "Less-than” cumulation 
Acreage Number Acreage Number 
Less than 3 20,350 Less than 3 20,350 
3, less than 10 268,422 i WS 288,772 
10g sO 607,762 2 FXO) 796 ,534 
20505 om TO 1,503,734 “ " 60 2,800,268 
50,,, £100 (2,474,768 " 100%) 13,775,022 
reo A Ae ly fe) 1,449,659 » SaL75 5,224,680 
175, oo ee en 260 530,795 a PS 260 5-756,473 
260” ae 600 . 475 f692 S00 86,281,167 
S00; ae L000) 149,812 »" —* 1000 6,380,979 
1000 and over 67,387 Total 6,448,366 


Fig. 301. The ‘‘Less-Than’”’ Cumulation. 
341 


Tag pene 


342 CHARTS AND GRAPHS 


While it would be meaningless to cumulate a historical 
~ series backward, and we therefore cumulate historical series 
only 1 in one Aigecden: it 1s possible to cumulate a frequency 
series from either end of the series. If the cumulation begins 
at the lower end of the data, it is called a “less-than’’ cumula- 
tive, for the sub-total or progressive cumulative figure repre- 
sents the number of items having less than the maximum quali- 
fication of the last added group. If the cumulation begins at 
the upper end of the series it is called a “more-than’”’ cumula- 
tive, the sub-total or progressive cumulative figure representing 


SIZE OF FARMS 
United States 
1920 
(Source:= Census) 


Simple series "Moreethan" cumulation 
Acreage Number Acreage Number 
1000 and over 67,387 1000 and over 67,387 
500 and less than 1000 149,812 SOOl as Es 217,199 
260 " My " 600 475 ,692 260 sae”, 692,891 
e765: 4! "260 630,795 WET 1,223,686 
1co (" My UA eslhrd 1,449,659 aifTo EAE Fy 2,673,345 
One My "100 1,474,753 50 se 4,148,098 
Zo," S ake) 1,503 ,734 20 deed 5,651,832 
vor uy EEX 607,762 10 Lo Bah 6,159,594 
3 " i Ht 268,422 3 ~~ S 6,428,016 
Less than 3 20,350 Total 6,448,366 


Fig. 302. The ‘‘More-Than’’ Cumulation. 


the number of item having more than the minimum qualifica- 
tion of the last added group. It is often useful to prepare both 
the “‘more-than” and “‘less-than”’ cumulatives for a frequency 
series and to plot them both on the chart as well as the series 
itself. 

It is one of the great advantages of the ogive that by its 
means a frequency series may be graphically presented and 
analysed whether or not the groups of the series be uniform in 
size (group-range). There is no labor of calculating values for 
equivalent groups. All question of staircase curves likewise 
disappears, even discrete series, when cumulated, being prop- 
erly shown smoothed. It is another advantage that several 


OGIVES 343 


ogives can be easily shown together and compared upon the 
same chart. The various series need not have uniform and 
identical group intervals. The ogive is therefore the only 
feasible method of comparing frequency series which do not 
have the same groupings. A further benefit is that the many 
ogives will generally be found to intersect very little, so that 
the confusion which attends superimposed frequency curves 
is avoided by ogives. 

For analytical purposes it may not be amiss to note that 
the median of a series is shown by the intersection of the two 


EXPECTANCY OF LIFE 
For Adults without Tuberculosis 
Registration Area, United States. 
1910 


(Source:- L. I. Dublin: Cost of Tuberculosie) 


Years Average 
of After Lifetime 
Age 
20 46.6 
25 42.3 
30. 38.1 
35 33.9 
40 29.9 
45 26.0 
60 22.1 
65 18.5 
60 15.2)" 
65 12.1 
20 9.5 
9% 7.2 
60 6.4 
85 4.0 
90 2.9 
95 1.9 


Fig. 303. An Example of a Frequency Series (So-called) 
Which Cannot be Cumulated. 


ogives, the more-than and the less-than, for the series, or by 
the value of the abscissae at the intersection of the curve with 
the ordinate of half the height of the 100% ordinate; while the 
mode is shown by the portion of the ogive in which the slope 
is steepest. These are statistical rather than charting concep- 
tions. The median may be described as the middle or central 


344 CHARTS AND GRAPHS 


observation, and the mode as the most common observation.! 
We may also note that two minor variations of the two 
cumulatives obtain, which depend in part upon the plotting 


SIZE OF FARMS 
United States 
1920 
(Source:- United States Census) 
(Pigures show number of Farms containing more than and less than speoificd number of acres) 


o ° 
oo z no wo c-) tal = o 
HaOD o ro 
n6e85 2 8 8 a 2 
More than DORAD wh 2. Oy iS 5 
aon & u 
ne SURSA & cS) o c) 
oe a a ee Cae ee - . 
2 B wonvnne aw cal 
3a 
i) oa ° to ~ @ ° 
5 98833 a ® i 2 S 2 
= mewon Oo oc ~ Le . a 
Less than aS oo = oq a pa o o 
o 2 a2 
uN DD | is 0 oa 
ae & De SS x = = 
a 9 © © ° o © 
o J 
Number of Acros CPALS S & 8 
Ses a g & 


6,000,000 


5,000,000 i ara mee 


4,000,000 


Yumber of Farns 


3,000,000 


* 2,000,000 


1,660,000 


° 
100 


3 8 8 8 8 re} 
2 oy + rr ry & 
Number of Acres per Farm 


Fig. 304. Two Ogives Are Always Possible. 


800 
900 
1000 
Over 


and in part upon the nature of the data. These variations are, 
for the “less-than” cumulative, a “less than and including” 


1 Readings from the curve, for the median, decils, quartiles, or percentiles, when 
secured by interpolation from the curve and not from plotted points on the curve 
give values, but of course do not give cases. The median case, for example, can be 
found only by reference to the original data, and exists only if the total number of 
frequencies be odd. The median value, however, is the intersection of the curve with 
the 50 per cent abscissa, and is obtained with increasing accuracy as the frequency 
groups are taken smaller and smaller, and the curve itself plotted in greater detail. 


Corrected 


Uncorrected Cumulatives 


OGIVES 345 


S1ZE OF FAMILIES 
Number of Children of 1,000 Women 
(married at least 16 years and having at least one child each) 
British Peerage Statistics 
(Source:- Yulo, Theory of Statistics) 


5 o ) rr) rT) ” 
© gore 2 o ° + @ © aS me, es ~ o © o ° a o 
> a a o o rey © fy © 
bey 2 a 3 - 12 + 0m _ a is - 
3 
a 
E Les cee macs 8 " ” % 
8 ey a ° © o 0 o " a © a oO a 
6 rt) a oO é a = © oO z a rf) t a & a road 
a ) o © e o o o a o ro a 
‘ : Spon a onmied Lear aie gee) Be OG SR ANS 
- - Or More 5} 3 r9) & © rr) r) a a a 
e ° ° © © a re ) cal © =F +. a « ° 
‘ - =" @ a rs rs 3 o o a o a 
More than ; 3 © e o if) » a a a mis 
2 ° ° x 2 2 a 4 o © © © a 
a ‘ns i) a a o ° dd a o ” 3 o a a a 3 
Cr Lees a nx 3 + o & o C) Q a a a a a 3 
o 2 co °o _ o n a Loe a © 0 o a 
nr ° 
ha o- wo La a 3 So a —_ oO nm 3 o nD a o 
Lese than a a BS * © i o © a D> o ro) ro) oa 


1000 


Number of Mothers 


900 


800 


700 


600 


600 


400 


300 


200 


100 


o Lal NN La ?: “ © e ao a oOo 
Number of Children 


Fig. 305. Showing the Four Possible Cumulations For Point Data. 


To find the median or quartiles, etc., graphically, it is necessary to use the dotted 
line showing the averages of these cumulatives (7.¢., the “corrected cumulatives’’). 


346 CHARTS AND GRAPHS 


cumulative (stated concisely as ‘and less’’) and, for the 
: eae 

“more-than”’ cumulative, a ‘more than and including” cumu- 

lative (stated concisely as ‘and more’). They are essen- 


HOURS OF LABOR 
Number of Wage-earners Employed in Manufactures 
According to Prevailing Hours of Labor 
United States 
1914 
(Source:- Statistical Abstract) 


28 More 61.86 16.35 1.68 
a3 
oes 
Ed ese 38.15 83.65 98.45 
bi 

”. or Nore”. 14.8 26.9 2.3 
uv es 
3s "More than” 88.2 48.9 5.8 0.68 
2% 
be 
3g "- or Less” 11.8 61.1 94.2 99.2 
Po 


“Less than -" 
ot 


80 


SO 


50 


40 


Percent of Workers 


30 


20 


10 


ous: x 
Under 48 64 60 72 Over 
Number of Hours 


Fig. 306. The Four Possible Cumulations for (Point-and-) Period Data. 


To find the median or quartiles, etc., graphically, it is necessary to use the dotted 
lines of the averages between these cumulations (1.c., the “corrected cumulatives’’), 


tially due to differences in the plotting points of data, and are 
sometimes more suitable for cumulations of discrete data. 

It has not generally been observed that the ogive is really 
simpler in its nature than the frequency curve. Because we 
have secured the frequency series by a very careful grouping 


OGIVES en «| 


HOURS OF LABOR 
umber of Wage-earners Employed in Manufactures 
According to Prevailing. Hours of Labor 
United States 
} 1914 ; { 
, (Source:- Statistical Abstract) 


| Percent of Wage-earnere 


Working more than and Se % e 
Including each Specified ze 8 — 
‘Number of Hours 
Percent of Wage-oearners o ee) | 
Working less than and a a : 3 
including each Specified ae ee < 
Number of Hours 
Percent of Wage-earnere bi tiene ate & 
Working each Specified a eee wun bi C4 
Number of Houre : 
See ad 
s a x s 
und f 5 ; 3 : 
lunber 
of Roure q 4 § s e 
ee als eg oe 
SR SRS eo ome 


S| 


ia 
LLIN 


Sy 


a 
f 


at 


LT 


& 
° 
= 
Go 


Fig. 307. The Rounded Ogives. 
Showing that median and quartiles, etc., gannot be easily found from two un- 
corrected ogives. 


of our original data, arranged in order of magnitude, and have 
then derived the ogive data from the frequency series by cumu- 
lation, we are prone to think of the ogive data as a somewhat 
more advanced and possibly more puzzling form of statistical 
series. The fact is, however, that the cumulation merely 
brings about a reversion to the original data in order of mag- 
nitudes, somewhat condensed as a result of the groupings. If 
we lay off the original data in the form of a bar-chart we will 
see at once that the ogive is merely a smoothed curve passing 
through the ends of the bars. It is for this reason that the size 
of groups or their uniformity is of no importance in making 
the ogive-chart, smaller groups merely defining the ogive-curve 
more precisely.” 


2 Cf. Robert E, Chaddock, in the American Statistical Association Quarterly, June 
1921, p. 769 ff. 


348 


Bridgeport 90.59 
Oakland 61 
Schenectady +66 
Camden 270 
Lawrence oT4 
Richmond 279 
Erie 289 
Lynn 291 
Elisabeth 998 
Columbus 1.04 
Reading 1,26 
Debver 1.29 
Worcester 1.30 
Rochester 1,30 
Tacoma 1.33 
Waterbury 1.34 
Antonio 1.36 
Jersey City 1,38 
Omaha 1.43 
New Orleene, 1.44 
Washington 1.46 
Cincinnats 1.47 
New Heaven 1.52 
Youngstown 1.55 : 
Hartford 1,54 
Kansas City, Kane 1,55 
Byracuse 1,58 
Yonkers 1.59 
Fall kiver 1.62 
Duluth 1.69 
Portiend, Ores 1.70 
Milwaukee 1.70 
St. Louis 1.80 
Dallas 1.81 
Dayton 1,95 
Lowell 1.96 
Los Angeles p98 
Newark 1.99 
Seattlo 2.01 
Fort Worth 2.06 
Louisville 2.08 
New Bedford 2,08 
New York City 2.08 
Spokehe 2.13 
dkron 2623 
Jacksonville 2.28 
St. Poul 2,30 
Minneapolis « 3,31 
Clevelerd 2.39 
Providence 2.41 
Chicugo 2.60 
Birmingham 2.65 
Des Moines 2.64 
Philadelphia 2.64 
Trenton 2.66 
Salt Lake City 2.67 
Nashville 2.71 
Springfield, Mesa 2,76 
Pitteburg 2.84 
Atlanta 2.85 
Patterson 3.02 
Boston 3.19 
Kaneas City, Mo. 3.21 
Scranton 3.35 
Indienapolis 3.56 
Oklehoma City 3.65 
Cambridge 3.73 
Memphie 3,92 
Baltimore 4,27 
Detroit 4.47 
Grand Repide 6.22 
Houston 6.78 
Toledo 6.82 
Norfolk 27,423 


CHARTS AND GRAPHS 


Dollars of Percapitea Love 
2.00 2.60 3,00 3.50 4.00 4.50 6.00 6.50 


FREQUSN CIES 9 us le 2 lo. 4 4 2 ° 1 CJ 0 2 ae 


20 


16 


10 


+50 1,00 1.50 2,00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6,00 
ollers of Ferceapits Lose 


Fig. 308. Evolution of the Ogive (Staircased). 


8.50 7.00 7,50 6,00 Over 


PERCAPITA FIRE LOSSES 349 
in 74 Large American Cities 
1919 


Qollers of Percapita Loss 
14 10 6 


2 
a 
ov 
a 
x 
é 
= 


ore than Cumlative 


= SE=S==5=== 

jeu SEES EESEEE=== 
== SSS =— = 

JES SSSSSaaaa= 

See Sages = 
a —\ QUARTILE — 135 | 4 = E 
aS 

== == = ze + 

| erate =o —— 
— 1S ai zai — 
== 


g $ & 
: att 4 ; 
| | 


ert TTL 
[| 
LL 

rH 


nn 
att 
i 


LAST|_QUARTILG — 2/79 ll 


15 


Hl 
lil 


iy 
LTT 


as Sa aee 
= SS 


5 arr 
© «50 1600 14650, 2.00 2450 3400 3450 4.00 4,50 5.00 5.50 6,00 6,60 7.00 7.50 8.00 Over 


8tmple Frequency 9 13 16 12 10 4 4 2 Co) 1 9 (0) UES 1 


oS =. ~ Z » 
° +50 1,00 1,60 2,00 2,60 3,500 3,60 4.00 4.60 5.00 6.50 6,00 6.50 7,00 7.50 8.0) Over 
Dollere of Percapita Loss 


Fig. 309, Evolution of the Ogive (Smoothed). 


350 CHARTS AND. GRAPHS 


SIZE OF FAMILIES 
Number of Children of 1,000 Women 
(married at least 15 years and having at least one child each) 
British Peerage Statistics 
(Source:- Yule, Theory of Statistics) 


Mothers having 
each specified 
Number of Children 


Mothers having up 
to and including 
each specified 
Number of Children 


Mothers having more 


63 


63 


67 


110 


947 


100 


210 


132 


342 


790 


140 


482 


658 


124 


606 


516 


113 


719 


394 


92 


811 


887 


52 


939 


113 


25 


964 


61 


22 


986 


36 


10 


996 


14 


998 


999 


1000 


i 


than and including 8 
each specified a 

Number of Children 
1000 


Number of Mothers 
bal 
8 


eth 
Baar) 


ws an o > oo a o a 
Number of Children per Mother 


Fig. 310. The Simple Curve and Its Two Ogives (Staircased). 


Where individual series are to be analyzed by themselves 
the horizontal scale for the ogive can be the same as the hori- 
zontal scale for the ordinary frequency curve from which the 
ogive has been derived and for which it has been substituted. 
The vertical scale, however, will have to be condensed so as to 
include the total of the entire series. But because a frequency 
chart is usually designed to show the comparative behavior of 
the phenomena studied, it is often useful to turn the actual 
data into percentages and to use on the chart a vertical scale 
calibrated in percentages. The percentage values are more 
useful for generalization and ready comparison with other 


OGIVES cas 


CURATION OF EMPLOYMENT 
Percent Listributicn of 1,306 Male ard 144 Female Employees on the Payrol) 
and 2,816 Male and 63 Fecale Ssparaticns who nad Served more than Specified Periods of Tine 
California Sugar Refinery 
Active:> May 31, 1918 
Separated = April 1, 1917 - May 31, 1918 
(Source:= Paul F. Brisserdon) 


© ate aD wo 
Cre Usles 8a5se © + EG a x to 
fF PEE AS = 
“A 
ad 
on 
o wo Lad 
ae Females Ssse + =} e Ms * 
£ oe - 
ODnHO wa hen} 
S53 Mle Sessa 8 vs , = 
~ DA ie! 
<s 90 
bw Ord 
se 
c= wo ~ 
Pen | Females 8825 ts iS bd 
CIS sea Sepa = 
ae 
w o 
Oe he & e © he 
eae te 5 5 5 
Pertod of Time oes & S ° ° © 2 
Fem a ca > - ~ > 
Onna ” ro) ov ca - wy 
00 


© 
gy 
Es 
Ss 
Q. 
a 
o 
E bal 
o 
o 
~ 
oD 
i 
polls Ato ast ee << on 
ts : = N 7 


Years 


Fig. 311. Relative Data. 


ogives, than the absolute scale figures of one particular series 
of observation or samples. For similar reasons you may find 
it desirable to turn these group-divisions which form the hori- 
zontal scale also into percentages, both in the data and on the 
chart. In both cases the percentages are percentages of the 
total or maximum limits of the series. When several frequency 
series are being compared, and the series differ both in the 
total number of observations or items in the series and in the 
group-divisions or group units into which the series is divided, 
this. little trick of turning all readings into percentages may 
be very useful, as by its means you can chart the ogives or 
cumulatives of all the series upon uniform chart-fields. The 
fields on which the curves are to be plotted should generally be 


352 CHARTS AND GRAPHS 


faces and Hours of Viomen 

Weekly Wage-rates and Hours of Labor of 3,720 Women 

in Deportment Stores and Dry-goods and Millinery Establishmento 
Virginia 
April 1, 1920 
(Source:+ Monthly Labor Review) 

(Figures show number of women receiving more than specified wages 

and working for more than specified number of hours.) 


a © AH oo» # 9 
wo 
Deily Hours La ee me 
a 
°o oS ce lsd o oO Nu Qa Oo a a Nn hod ao 
a = o ~~ £ ®@ © o oO 2 
Weekly Wages es OU ee Ol tanh eomcalaerins) aie ue CRON IO tes © a a 
” rn n rn rn o - ” n Led n N nN a 


2000 


1500 


Number of Women 


1000 


500 


Wages in Dollars per Week 


re 2 
- oa oa a a So 
Hours per Day 


Fig. 312. Comparison of Absolute Data is Sometimes Difficult. 


square, running from zero to one hundred per cent along both 
axes of the chart. Needless to say, the fields should be uni- 
formly positioned upon the sheets of paper so that the various 
charts to be compared may be freely subjected to “light analy- 
sis,” that is, to the method of analysis which consists of holding 
two or more charts together up to the light to detect the varia- 
tions of their curves. 

In the ogive chart we first meet with a type of chart which 
illustrates at the same time two different sets of figures for 
the same curve. There should therefore be space for data 


et 


OGIVES 353 


not only above the chart but to the right of the chart, and the 
chart field should not be placed close to or near to the right- 
hand margin of the paper as was the case in historical curves. 
The data above the chart is obviously the original data of the 
cumulative from which it is plotted, each data figure being 
placed above the ordinate or corresponding scale figure on the 


WAGES OF OFFICE, SALES, AND SHOP WORKERS (MALE) 
Weekly Rages of 932,808 Wage-earners, 63,519 Bookkeepers, Stenographers, 
and Office Clerks, and 23,756 Salesmen (not travelling) 
(All Males 18 years of age and over) 
Manufacturing Industries 
Ohio 
1919 
(Figures shew percentage of total receiving less than each specified wage) 
(Source: Industrixl Commission of Ohio) 


Sales 
Clerke 


63 

81 
1,13 
1.68 
2.6 


7.8 
61.3 
89.7 


oa ° 
° : 
2 x 
re) r 


39,9 


eae 
oats 


4.1 


222 

32 
43 
59 


Wage ° 
Earners 


1.0 


o) 
* 
c 


62.9 
" 


4 a 
2 ° 
o e 


34.9 


« 
u 

e 

= 


1.7 
34 


Office 2 
Workers 


+19 
33 
47 
61 
1.2 
2.6 
5.6 


” 0 ° 
° * . 
2 N - 
+ o = o 


29.7 


oS 
c 
~ 


10.9 


Percentage of Employees 


° wern Oo . ao cal wo fo} red 
a 7 a « x i) " 


° o 
” a 
Weekly Wage in Doliers 


Fig. 313. Comparison of Relative Data is Easy. 


x-axis of the chart. The data to the right of the chart field 
will be secondary or derived data, obtained by taking the 
readings or values of the curve at each of its intersections with 
the abscissae or horizontal rulings, using the corresponding 
scale figures of the y-axis for the new stubs, and taking the 
corresponding values along the x-axis as the new or derived 
data. This secondary data forms a new table of the same 
phenomenon rearranged so that the second variable or de- 


354 CHARTS AND GRAPHS 


pendent has become in a sense the independent one and its 
values would appear as the stubs in a retabulation. 

The ogive-chart is excellently adapted for the process of 
interpolation. The derived data just described are an example 
of this use of the chart. Interpolation, of course, is the name 
given to the process of reading new values between originally 
given values. Thus, by means of the ogive-chart, originally 
incomplete data can be filled out with interpolated figures. 
But the interpolated figures of course do not have the same 


WAOSS OF OFFICE WORKERS (FEMALE) 
Weekly Wages of 66,167 Adult Femsle Bookkeepers, Stenographers, and Office Clerke 
(1B Years of Age and Over 
Manufecturing Industries 
Onde 


(Pigures show percent of Total receiving less than each specified vage) 
(Source:+ Indvatrial Commission of Ohic) 


2R2S 2 4 Wego Liaite 
Comulation = aes co < 
i * (Beotle) ten peroent 
3 


” over $25.50 


7.19 11.89 
16.10 27.99 
13.92 86,24 

8.75 = 04,90 

2,96 97,06 

1.18 99.08. 


23.96 61.04 
20.38 12.32 


s 
Series Lied wre se cen ne 
” 


Over $25.50 


22.00 = 26.60 


20.60 = 22.00 


19,00 = 20,60 


17.75 + 19.00 


16,60 ~ 17.76 


Percent of Foren 


16.25 - 18.50 


13.60 + 16.26 


11.60 + 13,60 


Under $11.50 


Ss 2 
Weekly Wages 


Fig. 314. Secondary Data at the Right. 


degree of accuracy as the original data, being made on the 
theory that the line drawn between the plotted points of the 
original data has been correctly drawn. In spite of their pos- 
sible inaccuracies, they are often extremely useful in reducing 
otherwise incomparable frequency series to comparable group 
units or to uniform percentages. Where the chart itself is to 
be used, the interpolation is not necessary, the connecting 
lines which form the curve being in themselves plottings of 
interpolated values. But when the chart will not be presented 
in the final report or summary of the case, the interpolated 
figures are necessary, the interpolated figures being obtained 
from the chart before the chart itself is discarded. ? 


OGIVES 355. 


The ogive-chart is one of the most important and generally 
useful of the non-historical chart-forms and we will have oc- 
casion to return to it in the future with various elaborations 
and improvements.’ 


* Tt is thoroughly regrettable that statistical practice has so consistently considered 
the range of a frequency series tc be the independent variable and its frequencies the 
dependent. For while this is logical enough in the simple curve, it introduces into the 
ogive or cumulated curve, absurdities which make the latter not only unnecessarily 
obscure to the layman, but also brings about an unjustifiable violation of the primary 
rule in curve charting that the plot of all points should be independent x, dependent y. 

In the simple frequency curve, we are interested only in frequencies and they are 
obviously and properly dependent. The reduction ad initium of this chart to a bar- 
chart would, as has been seen, require the use of vertical bars, and the curve is but 
the short-hand connection of the tops of these bars. 

A very different case 1s that of the cumulated series. Here the frequencies are 
only in an immediate sense dependent; in the last analysis, they are independent and 
the range is dependent. As has been seen, the return of this curve to a bar-chart 
would require the use of horizontal bars—a fact which clearly illustrates the really 
dependent nature of the range. 

Statisticians have, however, so short-sightedly adopted the ogive as a derived 
chart (by cumulation) from the simple frequency curve, that they have followed its 
arrangement of the variables on the chart; and as the ogive has been confined to 
use by statisticians, the practice has become so settled that to advocate a change at 
this time would seem only to be adding confusion to a science which, more than all 
else, is in need of standardization. 

Life does not, however, always confine itself within academic rules, the round- 
about path must always be supplemented with a fence; and we venture the prediction 
that in time, as the ogive comes into commercial use, this arrangement will be scrapped 
and the ogive plotted on x-frequencies, y-range, passing first through a period of 
confusion which we do not seek to bring about. But he who must tell his story clearly 
or not at all will drop the old arrangement. A step in this direction, though perhaps 
unconcious, is that of the publishers of probabilities paper for ogives, who probably 
only by accident or for convenience have calibrated their scales so as to give 
x-frequencies, y-range. The chart in this form is more intelligible to the layman. 


CHarrer: XXX 
LORENZ CURVES 


Neither the frequency curve nor its ogive have that pecu- 
liar tang of popularity, that engaging frankness which appeals 
to the “average man.” If we consider the ogive the simpler of 
the two, since it is merely a curve Passing through the ends 
of horizontal bars, then indeed it is a curiously unscientific 
chart, in which the usual position of the dependent and inde- 
pendent variables is reversed. The frequency curve is then a 
short-hand method of arraying these, with a large degree of 
chance in its formation when various groupings have different 
results. And if we consider the ogive as a cumulated frequency 
curve it then involves all the obscurities of the simple frequency 
curve, augmented by further complexities of its own. We 


OUTPUT OF FACTORIES 


Wumber of Manufacturing Establishments 
of specified sizes 
(size being measured by value of preducts) 
United States 
1914 
Source; --U.S.Census 


Specified Number 
Value of Products of all Establishments 
having specified value of products 


per 
Establishment. 


Less than $5,000 

$5,000 = $20,000 
$20,000 ~- $100,000 
$100,000-$1,000,000 


$1,000,000 and over 


TOTAL 


Fig. 315. The First Measure—By Count of Items. 
356 


oS 


— 


LORENZ CURVES 357 


now take up, therefore, a curve which has more popular ele- 
ments in it, with a consequent sacrifice of statistical detail. 


OUTPUT OF FACTORIES 


Value of, Products of Menufacturing Establishments 
of specified sizes 
(size being measured by value of prodgots) 
United States 
1914 
Source: -- U.S.Census 


Speoified Value of Products 
Value of Products of all Establishments 
per having specified value of products 


Establishment 


Less than $5,000 233,381,081 


$5,000 = $20,000 905,693,168 


$20,000 - $100,000 2,550, 229,411 
$100, 000-31,000,000 8, 763,070,135 
$1,000,000 and over 11,794,060,929 


TOTAL 24,246,454, 724 


Fig. 316. The Second Measure—By Count of Units. 


All frequency distributions afford two possible series for 
precisely the same data. The first and more usual series is the 
count of items in each group of the distribution, the second 


OUTPUT OF FACTORIES 
Number and Value~of-Products of Manufacturing Establishments 
of specified sizes 
(size being measured by value of products) 
United States 


1914 
Source:- U.S.Census 


Specified All establishments having specified value-of-products 


Value-of-Products 
Establishment 


Less than $5,000 97,061 233,381,081 
$5,000 - $20,000 87,931 905,693, 168 
$20,000 - $100,000 56,614 2,550,229,411 
$100,000-$1,000,000 30,166 8,765,070, 135 


$1,000,000 and over 3,819 11,794,060,929 


TOTAL 275,791 2A, 246,434, 7% 


Fig. 317. Both Measures. 


358 CHARTS AND GRAPHS 


and alternative series is the count of the units of measurement 
attributed to these items. Thus a classification of farms by 
their size (in acres of land) can show us either the number of 
farms of each size or the aggregate number of acres in the 
farms of each size. The data of cities classed by their popula- 
tion may count either the number of cities of each specified 
number of inhabitants, or it may count the inhabitants residing 
in these cities. The Census of the United States, in its analysis 
of the manufacturing establishments of the country, according 


OUTPUT OF FACTORIES 


Number and Value-of-Products of Manufacturing Establishments 
of specified size 
(size being measured by value-of-products) 
United States 
1914 
(Noter-- All data in percentages of total or aggregate) 
Source:-- U. S. Census 


All Establishments having 
specified value-of-products 


Value-of-Produots 


Specified 
Value-of-Products 
per 
Establishment 


Less than $5,000 


$5,000 = $20,000 


Less than $20,000 


$20,000 + $100,000 


Less than $100,000 


$100, 000-$1, 000,000 


Less than $1,000,000 


$1,000,000 ana over 


Any value whatever 


Fig. 318. Cumulating the Percentages. 


to their employees, gives both the number of establishments 
and the aggregate number of employees in such establishments; 
in its analysis by value of products, gives both the number of 
establishments and their aggregate value of products. Ex- 
amples might be multiplied without end, for whenever we dis- 
tribute the items of any phenomenon into groups upon the 
basis of some units for measurement, we are then at liberty to 
count either the items themselves or their units of measure- 
ment, group by group. 


LORENZ CURVES an, Bho 


The thought, therefore, occurs to us that a chart could be 
made in which one of these series serves as the independent 
variable for the other, and in which the two values for each 


OUTPUT OF FACTORIES 


Value-of-Products and Number-of-Establishments 
of Manufacturing Establishments of specified size 
(size being measured by value-of-products) 
United States 
1914 
(Note:- All data in percentages of total) 
Source:- U.S,.Census 


Specified Size Establishments having 
as shown by specified value-of-products 
Valuo-of-Products 
per Number-of- Value~pf= 
Establishment Establishments Products 


Less than $5,000 


Less than $20,000 
Less than $100,000 
Less than $1,000,000 


Any value whatever 


Fig. 319. Data For the Lorenz Curve. 


group of items are made the co-ordinates of plotted points. 
Obviously, if we are not to have the curve which connects 


OUTPUT OF FACTORIES 


The Value of Produots of 
6pecified groups of Manufacturing Establishments 
United States '‘ 
1914 
In percentage figures 
(Notes=- All groups composed of establishments having 
least value of products.) 
Source:-- U & Oensus 


Number of Value of 
Establishments Products 


Percent Percent. 


Fig. 320. Data to Plot the Lorenz Curve. 


these points moving backward and forward as well as up and 
down, we must cumulate the series which is to be used as the 


360 CHARTS AND GRAPHS 


independent variable, and since it will be found that the curve 
has greater significance when both are cumulated, we in- 
variably cumulate both series. It is then a matter of indiffer- 


OUTPUT OF FACTORIES 


The Value of Products of 

specified groups of Manufacturing Establishments 

United States 

1914 

In percentages 

(Note:== All groups composed oumlatively of the 
establishments having least value of producta) 
Souroce;-- U. §. Census 


Percentage of aggregate "Value of Products” 


(0) — 
o 10 20 30 40 6 60 70 80 90 100 
Percentage of aggregate number of Establishments 


Fig. 321. The Lorenz Curve. 


ence which series be used as the independent one, and either 
series may be plotted upon either axis of the chart. It will be 
seen that the chart is closely related to the ogive, since it uses 
OUTPUT OF FACTORIES 
United States 
1914. 


The largest 10 percent of the factories preduce 78 percent of the values; the smallest 90, only 22. 
" " " " fn ) " " a on: « " 


20 2 87 
" i] 30 n ” ] " " 93 " " " ” : " ” a " Be 
f " 40 a " " 96 ” " " . . " 60 ” 4 
" " half " n " " 97 " " " a . " " ae s. 
n A] 60 i] “ . 98 i] " f : " n "40 " 2. 
F . 
P Re peta i PN ee eee SID a oe 2 $0, about 1. 
80 gos" bode ee ae " 20, Ces ke 
© " 90 f “ CT " " 908 “ e . a 8 in 10 . é 
H ’ 7 
(Noto:--"Large” and "small" refer to the sige of the factory as measured by the value of its products.) 
Also 
10 percent of the val d h 3 5 : 
. perc of the values are pro uced by the largest . percent of Factor ies; . by the smallest 908 
’ . 


OLOe vrerevee 


Fig. 322, What the Lorenz Curve Tells the Layman. 


LORENZ CURVES 361 


a cumulated series; it will also be seen that the chart omits 
altogether the classes or groups in which the data has been 
collected. Lastly, to produce a uniformity of these charts, and 
to facilitate the comparison of different distributions upon the 
same chart, all items are turned into percentages of the totals, 
and plotted upon percentage scales along both axes. 

When this type of curve is drawn upon a square field, with 
equal percentage scales upon each axis, it takes the shape of 
an archer’s bow, and the curvature of the bow has a peculiar 


LORENZ CURVE SHOWING THE DISTRIBUTION OF 
INCOMES IN 1918. 


PERCENTAGE OF TOTAL INCOME, 


10 20 30 40 50 Oo ‘O 60 90 
3 PERCENTAGE OF PERSONS BEGINNING WITH THE POOREST. 
From “Income in the United States,’’ by the National Bureau of Economic Research, by permission. 


Fig. 323. The Familiar Example. 


significance as an index of dispersion in the original distribu- 
tion. For a little thought will show that a uniform distribu- 
tion in which all items are alike will yield not a curve, but a 
straight line. The first ten per cent of the “population” in the 
series of personal incomes, for example, if all incomes were 
equal, would have ten per cent of the total income of the 
country, the first twenty per cent would have twenty per cent 
of the total income, and so on. Hence, the distance between 
the curve and the straight-line diagonal indicates the degree 
in which the series is removed from a perfectly uniform dis- 


362 CHARTS AND GRAPHS 


tribution—a feature which statisticians call dispersion or 
scatteration. 

The Lorenz curve, as this form of chart has come to be 
known, has not been much used except in the analysis of in- 
come and wealth distribution, but it is obvious that it is 
capable of use for any and all frequency series. It is simple 


OUTPUT OF FACTORIES 


The Value of Products of 
specified groups of Manufacturing Establishments 
United States 


1914 
In percentage 
Source:-- U. S. Census 


8 


N BeBe 
CORSEEEIES 


Percentage of aggregate "Gross Value of Products” 


© 10 2 30 40 50 60 70 80 90 100 
Percentage of aggregate number of Establishments 


Fig. 324. Two Curves of the Same Data By Using Both “‘More-than”’ and 
‘‘Less-than’’ Cumulatives. 


and popular in its appeal, without being in the least inaccurate 
or meaningless. It has certain advantages in the emphasis it 
throws upon dispersion and unequal distributions. Its chief 
disadvantage is in the omission of the group-by-group data 
for the series it illustrates, but this data is more in the nature 
statistical detail, and does not belong to what may be called 
a summary analysis of a distribution; to the average man such 
detail is confusing rather than helpful, while the results of the 


LORENZ CURVES 363 
dispersion, which this chart shows, form in his mind the meat 
of the matter.! 

The principles of the Lorenz curve can, however, be ex- 
tended to innumerable comparisons between frequency series. 
It is not necessary that the two series compared be the two 
alternative forms for the same data. Though the latter is 
usually the sounder practice, there may be occasion to bring 


OUTPUT OF FACTORIES 


The Value of Products 
of specified groupssof Manufacturing Establishnents 
and of specified groups of Employees therein 
United States 
1914 
In percentages 
Source: U. S. Consus 


a 
BESNe 
EIN 


3 
oO 


= Pee 


6 ors 


“a 
° 


as] 
fe] 


X 


ES 


Site 


rl 


VaR 


© 10 20 30 40 50 60 70 80 90 100 
Percentage of aggregate Number of. Establishments 
Percentage of aggregaté Number of Emplcyees 


Fig. 325. Two Curves of Different Data. 


Percentage of aggregate "Value of Products” 
ry 
3) 


Ne is 
ep ees 


together series which, for example, have different units of 
measurement. Thus the manufacturing establishments of the 
country are classified in the census as to number of em- 
ployees, value of products, value added by manufacture, horse- 
power used, and the like. Taking any one of these classifica- 


1 As no data can be easily appended to the Lorenz curve, unless we elect to give 
readings of the curve at various points (in which case data belongs at both the top and 
on the right side of the chart, to give readings for both variable scales), it is generally 
sufficient to append to the Lorenz curve the percentage cumulations from which the 
curve has been drawn. These afford to the inquisitive full details for the group-by- 
group distributions which the chart itself does not show. 


364 CHARTS AND GRAPHS 


tions, the census gives the other features just mentioned, for 
the establishments forming each group in the classification. 
The true Lorenz curve will then bring together, for example, 
the number of establishments having a specified value of prod- 
ucts and the group value of the products of these establish- 
ments. If we like, however, we can bring together the value 
of products of each group and the number of employees at- 
tached thereto, or the value added thereby, or any other 


OUTPUT OF FACTORIES 


"Value of Products” and "Value Added by Marwfacture" 
of specified groups of Manufacturing Establishments 
and of specified groups of Employees therein 

United States 


1914 
In percentages 
Source: U. S. Census 


LETS 
: y) gue ae : 
RE yas 
OOE ESO \ 


0 10 20 30 40 50 60 70 86 90 100 
Peroentages of aggregate 


Fig. 326. The Logical Form is Triangular. 


feature we desire to show. This forms a pseudo-Lorenz curve 
which, though slightly more complicated in principle, has the 
same popular features. 

Popularity is the main feature of the Lorenz curve, but 
it is not without its scientific significance. As already re- 
marked, the deviation of the curve from a straight line shows 
the dispersion or scatteration of a series. For this reason, it 
would seem useful to plot the Lorenz curve upon triangular 
or tri-axial ordinates, by omitting the useless half of the square 
chart-field, and to make the stright-line diagonal the base of 
the chart in order to emphasize the deviation of the curve 
from the straight-line diagonal. A second feature of the 


—— 


LORENZ CURVES 365 


Lorenz curve is that the lack of similarity between the two 
terminal parts or “‘tails’’ of the curve, indicates what statis- 
ticians call skewness in the data. In these ways, this form of 
chart is a useful tool for the technician, and yields an intelli- 
gible message of details in which he is interested and which 
will escape the lay reader. Primarily, however, the Lorenz 
curve is popular-——the one and only way of making an inter- 
esting picture of a frequency distribution for the average man 
and of bringing strongly home to him the practical aspects 
of the frequency distribution presented. 


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CHAPTER XXXI 
THE GENEALOGY OF NUMBERS 


~The first mathematical operation in the world was probably 
more difficult for its discoverer, than the most complicated 
mathematical processes are for us today. Prehistoric man was 
able to master that initial operation and hence you are con- 
fronted with the disconcerting question, ‘‘are you intellectually 
weaker than the cave man, the stone-age man and the iron-age 
man?” If you admit the charge, or if you have already mas- 
tered higher mathematics, you should skip this chapter and 
continue in the straight and narrow road of charting, for in the 
first case you will not understand it and in the second you 
will not need it. The chapter is by way of being a comic inter- 
lude in which the reader is invited to wander down a by-way 
into pure mathematics, which will give him a theoretical 
understanding of the charts which follow. For a practical 
working knowledge of them, this theoretical understanding 
is not needed, but it will be a source of abiding satisfaction to 
him and will incidentally raise his batting average against 
charting errors and mistakes. 
The first mathematical operation was that of counting off 
or numbering. That is to say, standing in the middle of a 
road, you walked forward and your first step was your “first”’ 
step, your next was your “second” step, your next was your 
“third” step, and so on. The implements are called ordinals 
(“first,” “second,” “third,” etc.). The result of this counting 
off or numbering is to give you the number of steps you have 
taken, that is, measurement or mensuration. The measure- 
ment comes in the form of what is called a cardinal number 
“one,” “two,” “three,” and so on). Hence note that you 
have the following: 
Materials: Distinct items 
Operation: Counting off or numbering 
Result: Ordinal numbers 


366 


THE GENEALOGY OF NUMBERS 367 


Materials: Ordinals 
Operation: Measurement or mensuration 
Result: Cardinal numbers 


Several centuries may have passed before another bright 
young chap came along who was a little less hairy than his 
ancestors and had a little higher forehead. He discovered 
that if he walked five miles one day and five the next, it was 
the same as walking ten miles in all, the interesting thing being 
that the same two cardinals always made the same third car- 
dinal. This made possible the great and fundamental law 
upon which our civilization is said to rest, that two and two 
make four. The operation is known as “addition.” It is a 
sort of multiple measuring. Its result is a “sum.” An in- 
verse operation exists which is called “‘subtraction,” the result 
of which is called a “difference.” In this inverse operation we 
first meet with what are called ‘“‘negative numbers” giving 
rise to a conception that numbers can sometimes be either 
“positive” or “‘negative.” And in this inverse operation we 
must distinguish the two numbers operated on, calling the first 
the ‘‘minuend”’ and the second the “subtrahend.” Now note 
that you have the following: 


Direct Inverse 
Materials: Terms Terms 
Ist material —_—_——— Minuend 
2nd material Increment Subtrahend 
Operation: Addition Subtraction 
Result: Sum Difference 


Again many centuries elapse before the third act. This 
time the inventor discovers that if he walks five miles every 
day for five days, it is the same as walking twenty-five miles 
in all, and that no matter how often he repeats the operation 
the result will always be the same for each two given numbers. 
From this we discover the law that two times two make four, 
and with it the multiplication table. The process is called 
multiplication and is a sort of multiple adding. Most of our 
so-called multiplying machines are built on this principle, the 
operator simply turning the crank of the machine often enough 
to add the quantity the right number of times. The reverse 
process is called ‘division’ and is a sort of multiple subtrac- 
tion. The result of “multiplication” is called a “product,” of 
division a “quotient.” And the reverse process, division, gives 


368 CHARTS AND GRAPHS 


rise to a new type of number called a “fraction,” showing us 
that numbers can sometimes be “integrals” or whole numbers 
and sometimes “fractions” or part numbers. And remembering 
the negative numbers we met, we also find negative products 
and quotients or fractions. Note now that we have the fol- 
lowing: 


Direct Inverse 

Materials: Factors Factors 
Ist material Multiplicand Dividend 

2nd material Multiplier Divisor 
Operation: © Multiplication Division 
Result: Product Quotient 


Again a long time passes. The fourth act begins. Some 
one says: A five-mile distance walked by five persons, total 
walking, 25 miles (in which 5 is used twice as a factor), on 5 
different days, total 125 miles (in which 5 is used three times 
as a factor), in five different cities, total 625 miles (in which 5 
is used four times as a factor) and in five different countries, 
total 3125 miles (in which 5 is used 5 times as a factor). Now, 
says he, instead of writing “5 X5X5X5X5” why not write 55 
and be done with it. His invention, you see, is clearly one of 
notation. His process is called “‘raising to a power,” his result 
being a “power” of the original number. It is a sort of multiple 
multiplication. The inverse operation is called “reducing to 
a root” the result being a root of the original number. It is a 
sort of multiple division. We use the method when we say 
that the square (or second power) of two is four. The little 
number up in the corner is called the exponent. When it is in 
the righthand corner it signifies raising to a power, and when 
on the lefthand side in a radical sign it signifies the inverse 
operation of extracting a root. Again the raising to a frac- 
tional power also signifies the inverse process. And in the in- 
verse process we meet with the square root of negative num- 
bers, which we call “‘surds” or “irrational numbers.” And we 
also find negative exponents. Now note that you have: 


Direct Inverse | 
Operation: Involution Evolution 
Materials: Ist Number Number 
2nd Exponent Fractional exponent 


Result: Power Root 


THE GENEALOGY OF NUMBERS 369 


Here, the author, in the role of stage manager, must step 
out in front of the curtain, with a little speech of apology. 
The play would progress better had not the playwrights, that 
is the mathmeticians, been badly put to it to find new names 
and symbols for their operations. They have progressed 
bravely up to this point. Thus reviewing their work you find: 
Counting off: \st, 2nd, 3rd, 4th, 5th, etc., gives us Measurement; 
Peau dit by ds CC: 

Multiple Measuring: 1, 2, 3, 4, 5, and 1, 2, 3, 4, 5, gives us 
Addition; 5 +5 =10. 

Multiple Addition: 54+5+5+5+5 gives us Multiplication; 
5X5 =25. 

Multiple Multiplication: 5X55 5x5 gives us an Involu- 
prone 553.125, 

So far, they have given us a marvelous system for the easy 
notation of their ideas. You will notice that the phrase 55 is 
a highly compressed expression, which would otherwise have 
to be written5 x55 .. (to 5 times) or 545454545... 
(to 625 times) or 1+1+1+1+1 ... (to 3125 times). But 
we warn you that this simplicity is at an end. Examine the ex- 
pression, 3125 =55. Substitute for it the general algebraic ex- 
pression 4=B°. This describes 4 as the C-th power of B. 
From it you can readily derive the expression for B, as follows: 


B= VA, that is B is the C-th root of 4. But they have no 
_ convenient symbol for C, the exponent of the power to which 
‘B must be raised to equal 4, and they can only give you a 
cumbersome word by which you can describe C as—but wait 
and see. 

The fifth act, with which, so far as we are concerned, the 
play should end happily, opens with a young man who discovers 
that the fifth power of five is the same thing as multiplying to- 
gether the second and third powers of five, and that, in general, 
to multiply two powers of the same number together you need 
merely add their exponents, thus B’ x B’=B™, Likewise, 
to divide a power by another power of the same number, you 
need merely subtract their exponents, thus B’ + B? = BY. 
Whereupon he promptly says, let us change all numbers in the 
world into powers of one common and universal base number 
and then we shall be able to substitute for the lengthy tedious 
process of multiplication and division, the simple and easy 
process of addition and subtraction. Instead of multiplying 


379 CHARTS AND GRAPHS 


together a long series of large numbers we would only need 
to add their corresponding exponents of this universal base. 
The discovery was in the nature of a miracle. To add expo- 
nents instead of multiplying powers! The process has been 
accounted one of the nine wonders of the world. And it’s as 
easy as falling off a log! ‘ 


Then this excellent discoverer has to spoil his work by 
using a long and terrifying name for his process. For he calls 
it logarithmation. He calls his exponents of the common base, 
logarithms to that base. He calls his table of exponents a table of 
logarithms. He cannot think up a new symbol and when 
asked what c is in the equation, 4 = B’, he writes: 


c=log , A 


(This is read, “‘c equals log 4 to base B’’.) For he abbreviates 
his long word logarithm by the short word Jog. But it is no 
use. The public has decided that the use of logarithms is not 
as easy as falling off a log. All of which goes to show that 
there is something in a name after all. The public fell off the 
logs long ago and has been off them ever since. After this 
unhappy denouement, we introduce the following pageant, 
as additional entertainment to an audience which has sat 
faithfully through five tedious educational acts.! 


Marshal before your eyes the countless myriads of num- 
bers known to man. (For the sake of simplicity consider the 


whole numbers only, forgetting for the moment the fractions.) 


Arranged in single file from zero out into infinity, it would 
take forever for their procession to pass. For there is literally 
no end to them. Marching by at the rate of one at every 
tick of the clock, the first ten thousand would pass in an hour. 
And marching day and night, after four days one million would 
appear. But it would be ten years before the first number of 
ten digits, the first billion number, comes into view. And as 
to the trillion, that would not yet have appeared if the parade 
had begun before the pyramids were built. Yet the trillion 
is no longer a stranger to financial circles and is a poor small 
thing in the world of science. 


Now hovering over the shoulder of every one of these num- 
bers the close observer might discover its spiritual counterpart, 
its soul. Subject this soul to close analysis and you will find 


1 The foregoing text has been largely modelled after the excellent introductory 
chapter in “Engineering Mathematics” by Charles P. Steinmetz. 


a 


THE GENEALOGY OF NUMBERS Bi 


it is the exponent which will raise some common universal 
base-number to the value of the number itself, and since our 
numbers are arranged on the decimal system, the most con- 
venient base figure for the exponents or souls is the number 
ten. From this we may easily identify the souls of all powers 
of ten. Thus it is easy to see that the soul of ten itself is 1, 
since ten is the first power of ten. It is easy to see that the 
soul of one hundred is 2, since it is the second power of ten 
(that is, the base number, ten, must be taken twice as a factor 
to give us the number one hundred). It is easy to see that the 
soul of one thousand is 3; that the soul of one million, for ex- 
ample, is 6; and so on. Indeed we quickly discover that every 
whole or integral soul belongs to an even power of ten and coin- 
cides with the number of ciphers between the initial digit one 
and the decimal point. In short, the soul tells us the position 
of the decimal point. 


Going back into small numbers and fractions it is equally 
simple. The soul of one, for instance, is 0, for the zero power 
of any number, including the base ten, is one. Notice that the 
soul still tells us the position of the decimal point, for the 
latter is immediately beside the initial digit, without any inter- 
vening digits. Now what is the soul of one tenth? Obviously 
it is —1, for as you know, to convert a denominator into a 
numerator we need merely change the sign of its exponent, and 


1 es eee igs 
10 =10'. Likewise the soul of one-hundredth is —2, for 100 
1 


102 = 10": The soul of one-thousandth is —3, of one ten- 


thousandth is —4, of one-millionth is —6, and so on. And if 
we write these fractions, one-tenth, one-hundredth, one-thou- 
sandth, one ten-thousandth, or one-millionth, as, respectively, 
ee Ol, 2001, 000,17’ or “*000,001,”> we ‘shall#see that 
their souls, namely —1, —2, —3, —4 or —6, tell us again the 
positions of the decimal points. Only this time, since the 
decimal point has been moved backwards, that is, to the left of 
its position beside the first significant digit (initial ciphers are 
not called significant), the soul has become negative. A 
quaint and convenient fact, that the soul always tells, by its 
sign and whole or integral part, the precise position of the 
decimal point in a number. 


372 CHARTS AND GRAPHS 


But what of the souls of numbers lying between the even 
powers of ten? There are eight numbers between one and ten, 
eighty-nine between ten and one hundred, and many, many 
more between the higher powers. They too have souls, but 
“it is clear that they cannot have even whole or integral souls, 
for these belong to the powers themselves. The numbers two, 
three, four and so on, for example, lie between one and ten; so 
they must have souls between 0 and 1. We easily conclude, 
therefore, that their souls must be fractions somewhere be- 
tween 0.0000 and 1.0000. That this is correct we can quickly 
demonstrate. Consider the square root of ten, a number, as you 
know, a little greater than three. It is obvious that its soul 
must be z since the square root of ten is the one-half power of 

ys Tae 

ten, that is V10=10%. Hence for the square root of ten we 
have a soul 0.5000, lying as you see between 0.0000 and 1.0000. 
Mathematicians have figured out to many places the souls of 
other numbers, that of three (which is but a little less than the 
square root of ten) being to four places 0.4998; that of two being 
0.3010. Remember these two and you will always be able to 
reconstruct the souls of almost all other numbers without as- 
sistance. In short, the souls of all numbers other than powers 
of ten are not integral, but are fractional. 


Returning to the parade, let us call a halt to the intermin- 
able thing and hold a grand review of the numbers, marshalling 
them according to their significant digits. In the entire bat- 
talion of numbers there will then be but nine regiments, each 
led by one of the significant digits, one, two, three, four, five, six, 
seven, eight, or nine. Each regiment will again be composed 
of ten companies in which the second digit is one of the ten 
numerals, zero to nine. Each company will be divided into ° 
ten platoons in which the third digit likewise varies, each 
platoon into ten squads whose fourth digits vary, and each 
squad will be similarly divided into ten subdivisions. This 
subdivision could proceed indefinitely. You will notice that we | 
have here disregarded entirely the position of the decimal 
point. Now the interesting thing about this arrangement is 
the fractional parts of the souls. For the souls would never 
repeat themselves in this review, but there would be one 
fractional part of a soul assigned to each file or succession of 
significant digits, and belonging to that particular file al- 


THE GENEALOGY OF NUMBERS 373 


ways, regardless of changes in the position of the decimal 
point. The integral parts of the souls would indeed change 
with the change of the decimal point, but not the fraction. 
Thus glancing down the file “200000,” we find that the soul 
of two (2,00000”) is 0.3010, that the soul of twenty 
(“20.0000’’) is 1.3010, that the soul of two hundred (200.000” 
is 2.3010, that the soul of two thousand (‘2,000.00’’) is 3.3010, 
and so on. In short, while the integral parts of souls are the 
same for similar positions of the decimal point, the frac- 
tional parts of the soul are the same for similar successions of 
(significant) digits in numbers. 

But this glimpse of the souls of numbers must come to a 
close. From now on in this book (and in all other books), you 
will meet them again only under the prosaic names of logs or 
logarithms, or more precisely, common or Briggsian logarithms.1 
When logarithms are used, the natural numbers for which 
they stand are sometimes called anti-logarithms. The frac- 
tional part of the logarithm is called the “mantissa” and be- 
cause it 1s the same for all similar combinations of natural 
numbers or significant digits, it forms the body of the table 
of common logarithms. The integral or whole part of the 
logarithm is called the “characteristic,” and since it records 
the position of the decimal point is not shown in logarithm 
tables but is left to be determined by inspection. With the 
information dispensed in this chapter, in as heavily sugar-coated 
pellets as we could provide, you are prepared to meet, master 
and make use of any logarithm which strays your way as if 
it had been your life-long servant—no, no, much better than 
that! 

In this chapter we shall go no further into the uses of 
logarithms. They shall sit up and perform for us through the 
major part of the rest of this book. We merely repeat, for 
your lasting remembrance (and don’t ever forget it) their 
fundamental relations: 

log 4+log B=log (4 XB) 
log A -log B=log (4+B) 
1 “Logarithms were invented and a table published in 1614 by John Napier of 


Scotland; but the kind now chiefly in use proposed by his contemporary, Henry 
Briggs, professor of geometry in Gresham College in London.”—Century Dictionary. 


CHARTS AND GRAPHS 


10 
11 
12 
13 
14 


AS 
16 
17 
18 
19 


20 
21 
22 
23 
24 


25 
26 
27 
28 
29 


30 
31 
32 
33 
34 


35 
36 
37 
38 
39 


40 
41 
42 
43 
44 

45 
46 
47 
48 
49 


50 
51 
52 
53 
54 


0212 
0607 
0969 
1303 
1614 


1903 
2175 
2430 
2672 
2900 


3118 
3324 
3522 
3711 
3892 


14065 
4232 
4393 
4548 
4698 


4843 
4983 
5119 
5250 
5378 


5441 5453 5465 5478 5490} 5502 
5563 5575 5587 5599 561115623 
5682 5694 5705 5717 572945740 
5798 5809 5821 5832 5843) 5855 
5911 5922 5933 5944 5955] 5966 


6021 6031 6042 6053 6064] 6075 


6128 6138 6149 6160 6170] 6180 
6232 6243 6253 6263 6274] 6284 
6335 6345 6355 6366 6375] 6385 
6435 6444 6454 6464 6474] 6484 


6532 6542 6551 6561 6571} 6580 
6628 6637 6646 6656 6665} 6675 
6721 6730 6739 6749 6758] 6767 
6812 6821 6830 6839 6848] 6857 
6902 6911 6920 6928 6937 | 6946 


6990 6998 7007 7016 7024] 7033 
7076 7084 7093 7101 7110} 7118 
7160 7168 7177 7185 71937202 
7243 7251 7259 7267 727517284 
7324 7332 7340 7348 7356|7364 


0253 
0645 
1004 
1335 
1644 


1931 
2201 
2455 
2695 
2923 


3139 
3345 
3541 
3729 
3909 


4082 
4249 
4409 
4564 
4713 


4857 
4997 
5132 
5263 
5391 
5514 
5635 
5752 
5866 
5977 


6085 
6191 
6294 
6395 
8493 


6590 
6684 
6776 
6866 
6955 


7042 
7126 
7210 
7292 
7372 


0294 
0682 
1038 
1367 
1673 


1959 
2227 
2480 
2718 
2945 


3160 
3365 
3560 
3747 
3927 


4099 
4265 
4425 
4579 
4728 


4871 
5011 
5145 
5276 
5403 


5527 
5647 
5763 
5877 


5988, 


6096 
6201 
6304 
6405 
6503 


"6599 


6693 
6785 
6875 
6964 


7050 


7135: 


7218 
7300 
7380 


0334 
0719 
1072 
1399 
1703 


1987 
2253 
2504 
2742 
2967 


3181 
3385 
3579 
3766 
3945 


4116 
4281 
4440 
4594 
4742 


4886 
5024 
5159 
5289 
5416 


5539 
5658 
5775 
5888 
59:99 


6107 
6212 
6314 
6415 
6613 


6609 
6702 
6794 
6884 
6972 


7059, 


7143 
7226 
7308 
7388 


Fig. 327. Table of Logarithms, 1-5. 


PoP. 


1.2 3.4 5 


4 812172) 
4 811-15 19 
37-1014 17 
3.61013 16 
3 6 (9:42 15 


3.6. 8-11 14 
3.5. 811 15 
2 5. 71012 
2.5. 7. 912 
24. 7-911 


2.4 6 811 
2.4 6 810 
2.4. 6. 810; 
24.6.7 9 
2. 5.7. 9| 


—-~ NNN WN 


eye egy loeweog 
aug ye 


aad) ot eid pee ah 
> + + 2 fb 
TH MAAR 
AaAnnn 


aan DARAADDAD 


eee 


aon 


| 


; 
prmnynp 


ats 


tp mp me ts 


56 
57 
58 
59 


60 
61 
62 
63 
64 


65 
66 
67 
68 
69 


7 
71 
72 
73 
74 


75 
78 
ae 
78 
79 
$0 
81 
82 
83 
84 


85 
86 
87 
88 
89 


90 
91 
92 
93 
94 


95 
96 
97 
98 


99 19956 


9031 
9085 
9138 
9191 
9243 


9294 
9345 
9395 
9445 
9494 


9542 
9590 
9638 
96895 
9731 


9777 
9823 
9868 
9912 


8500 
8561 
8621 
8681 
8739 


8797 
8854 
8910 
8965 
9020 


9074 
9128 
9180 
9232 
9284 


9335 
9385 
9435 
9484 
9533 


8506 
8567 
8627 
8686 
8745 


8802 
8859 
8915 


8971, 


9025 


9079 
9133 
9186 
9238 
9289 


9340 
9390 
9440 
9489 
9538 


CO w BO & week ww 


wWwwwiw 


Wwwwwnw 


Fig. 328. 


Table of Logarithms, 5-9. 


376 CHARTS AND GRAPHS 


A device which will do all this is worth knowing.? In the 
language of valedictorians, we commend to your early atten- 
tion a small table of logarithms and many pleasant hours of 
easier computing therewith. 


2 From logs, that is, the logarithms of numbers, it is but a simple step to proceed 
to loglogs, that is, the logarithms of logarithms. Consider the phrase 4%, in which 
A and B are any values we wish, such as 29.37 and 43.921. We can write log 42=B 
log A. This reduces the involution to a mere matter of multiplying B into the log 
of A. But if the multiplication be tedious, as with the values first instanced, it will 
be simpler to write 

log (log 43) =log (B log 4) 
=log B+loglog 4 
and proceed by addition. The loglog of a number is the logarithm of its logarithm, 
and is found in the log tables by treating the logarithm as a number. 


CHAPTER XXXII 
THE LAW OF ORGANIC GROWTH 


The law of organic growth, as it is called, is well-nigh as 
important to the practical business man as the law of cause 
and effect, but is unfortunately much less understood. The 
chart papers and methods discussed in this section of the 
book are designed to interpret statistics in the light of this law. 
To the uninitiated their construction remains a mystery, but 
to those who know the law which is the key to their meaning 
they are so valuable as to eclipse and almost to obviate all 
other chart methods. The law relates to the way in which a 
large majority of natural organic forces have been found to 
grow or change. It prescribes or defines the manner in which 
this growth or change will take place. The lawis that, at 
regular intervals of time, each new value will be a constant 
percentage of the immediately preceding value. 

This feature of a constant relation between successive 
items in a series marks what mathematicians call a progression. 
There are several kinds of progressions, only one of which 
follows the law of organic growth. By far the simplest form 
of progression is the one called arithmetical. In the arith- 
metical series or progression, each item differs from the pre- 
ceding item by a constant amount (quantity, difference or in- 
crement). The series progresses from item to item either by 
addition or by subtraction of this amount. For example, in 
the series, 1, 2, 3,4,5,6, . . ., the constant increment is +1. 
in the series 4, 3,2, 1,0, = 1,-—2, . ..., the constant 1s —1. In 
business, the familiar instance of the arithmetical progression 
is the accumulation of simple interest. 

Another and a very different series or progression is the one 
called geometrical. In a geometrical progression, each item 
differs from the preceding item by a constant ratio (rate, per- 
centage, factor, multiplier, or divisor). The series progresses 
from item to item by multiplication or division by this con- 


377 


378 CHARTS AND GRAPHS 


stant ratio. For example, in the series, 1, 2, 4, 8, 16, 32, 64, 

. ,the constant ratio or factor is 2. In the series 4, 2, 1, 
1/2/31, ... the constant is 3. In business, a familiar 
instance of the geometrical progression is the accumulation of 
compound interest. And it is the geometrical progression 
which the law of organic growth prescribes. 

It is interesting to study these two types of progression, 
for they are the gist of the distinction between the curve 
charts which we have so far considered and the curve charts 
to which we are coming. In the first place let us consider the 
relative speed of these progressions. Compare the series 1, 2, 
34,5, 6, 3. % withthe series, 1) 254; 85,16, 32564, fon jwand 
you will see that the geometric progression rapidly outruns 
the arithmetical one. These series have begun at unity and 
progressed by a 100% increase, which is a fairly rapid rate of 
increase. But the acceleration, from the arithmetical point 
of view, of the geometrical series will still be evident if we 
take a slower rate of increase such as 10%. Starting at unity, 
the arithmetical series will be 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 

. . . while the geometrical series will be 1, 1.1, 1.21, 1.33, 
1.47, 1.62, 1.78, 1.96. . . . Of course either of the two types 
of progressions can begin with any item and increase at any 
rate, but from any point you wish to choose, if the rates are 
the same for the two series, the geometrical progression will 
always increase more rapidly than the arithmetical one. 

On the other hand, in the decreasing or diminishing direc- 
tion, the arithmetical progression will leave the geometrical 
one behind. Compare the series, 2, 1,0, —1, -—2, -3,... 
with the series 2, 1, 4, 4, 3, ds, . . ., and this will be evi- 
dent to you. And here we come to an important distinction 
between the two series, namely that while the arithmetical 
series can reach zero and pass into negative values, the geo- 
metrical series can never reach zero at all. In the example 
just given, the arithmetical series diminishes by subtracting 
3 of the value of its first item and quickly passes zero, but 
the geometrical series diminishes by division by 2, and gives 
no indication of ever reaching the value of zero. Now we 
could have made the geometrical series begin with any other 
positive value and decrease at any other rate we please, but 
it still would be impossible for us to bring the geometrical 
progression down to zero. We can, by constantly diminishing 
it, that is, by repeatedly dividing its last item, bring it as 


Dake 7 


THE LAW OF ORGANIC GROWTH 379 


close to zero as we please, without ever succeeding in entirely 
wiping it away. In mathematical language, zero is the in- 
finitesimal limit of the geometrical progression. 

Malthus made popular the distinction between these two 
kinds of progression with his theory that while the population ~ 
of the world increased geometrically, the wealth of the world 
increased only arithmetically and hence soon limited the wel- 
fare of the population. But his theory has been proved to be 
false, the wealth of the world appearing to increase geomet- 
rically, though sometimes at a slower rate than the popula- 
tion. Indeed an attempt has recently been made ! to invest 
mankind with a peculiar and to some extent exclusive power 
of geometrical progression, both in mental and physical accom- 
plishments, a theory to which the work of Professor Ogburn? 
in suggesting something like a geometrical progression of all 
that may be called human civilization, bears important con- 
firmation. While it may not be so certain that the power of 
geometrical development is exclusively a property of human 
individuals, not shared by individual animals and plants, it 
is safe to say that human accomplishments, including business 
enterprises, are as often subject to the law of organic growth 
as are natural forces. 

The law of organic growth therefore is the proper criterion 
for the business man in judging the development of a business, 
as it is for the economist in his study of industrial and socio- 
logical records. In applying this criterion, we must forget 
(for the time being at least) the amount of increase in our 
business from year to year and center our attention upon its 
rate of increase. What are the year-to-year percentages of 
increase? If the 1911 sales were 10% larger than the 1910 
sales and the 1912 sales were again 10% greater than the 1911 
sales, the business has increased in accordance with the law of 
organic growth. If the 1913 sales were again 10% greater than 
those in 1912, the increase has still followed the law. If the 
1913 sales were only 8% greater than those in 1912, there 
has been from the point of view of the law, a definite slowing 
down, or falling off in the rate of growth, which must either 
be explained by conditions outside the control of the company, 
such as a general business depression, or is a harbinger cf ill 


1 Cf. Korzybski, Alfred, Manhood of Humanity, E. P. Dutton & Co., New York, 1921. 
2 Cf. Ogburn, William Fielding, Social Change, B. W. Huebsch, ae New York, 


1922. 


380 CHARTS AND GRAPHS 


omen which should call for almost as careful consideration by 
the directors as if there had been an absolute loss. On the 
other hand, if the 1913 sales were 15% greater than those in 
1912 and the event is not to be explained by forces outside 
of the control of the company, there is reason for far-sighted 
rejoicing and thanksgiving among the directors. 

That the results of the use of this criterion are radically 
different from the results reached by a study of the amount of 
change, is evident from the fact that the former may at times 
be directly contradictory to the latter. Let us suppose that 
in its first year the gross sales of the house amount to $50,000, 
and in the second year to $100,000, that is, there has been an 
increase of $50,000, or 100%. If in the third year, sales 
amount to $160,000, obviously the amount of increase has gone 
up from $50,000 to $60,000, but the rate of increase has fallen 
from 100% to 60%. If in the fourth year sales amount to 
$225,000, the amount of annual increase has again risen to 
$65,000 but the rate of annual increase has fallen to about 
40%. If in the fifth year sales amount to $300,000, the 
amount of annual increase has again risen, this time to $75,000, 
and the rate of annual increase has again fallen, this time to 
33%. This fictitious example makes clear how illusory would 
be any conclusion based wholly upon the amount of change 
from year to year and how important it is, that the annual 
rate of change should be watched, that is, that the records 
should be studied in the light of the law of organic growth. 

As a matter of fact, few business houses follow closely the 
law of organic growth for any considerable period of time. Or 
perhaps it would be better to say that though operating under 
the law of organic growth, they fail to maintain a constant rate 
of change. For it is certain that they operate under this law 
rather than under any law of arithmetical progression. Theo- 
retically, perhaps, given constantly similar external conditions 
and internal efficiency, the growth of a business house would 
conform to a geometrical series and illustrate perfectly the 
law. But as a matter of fact, the individual business house is 
at the mercy of a large number of external forces which lie 
outside of its control and do not remain constant but are ever 
changing, and the records of its growth therefore show a great 
amount of the play of what we might call chance variation. 
Business men are accustomed to thinking, in some fields, of a 
very definite saturation point intheir markets. Of course when 


THE LAW OF ORGANIC GROWTH 381 


such a point is approached, it becomes a limit which will neces- 
sitate a slowing up of the rate of increase, in spite of otherwise 
equal conditions, which would have favored a strict adherence 
to the geometrical progression. These individual variations 
and any approach to limiting points do not invalidate the law 
of organic growth, nor do they diminish its value as a criterion 
of business success. They are separate and additional forces 
imposed upon the development of the individual business, 
their co-action with the law of organic growth determining the 
fluctuating records of the house. 

In entire industries, or in large aggregates of individual 
business records, the adherence to the law of organic growth 
is much closer and the operation of the law easily seen. In 
the last two decades, the automotive industry has afforded a 
spectacular illustration of a geometrical progression with a 
very rapid rate of increase, although a decline in this rate in 
recent years is terrifying manufacturers with visions of the 
approach of a potential saturation point in their domestic 
market. Two other recent industries, whose entire history 
can be covered in the last twenty years, show similarly close 
adherence to the law, both the phonograph and the moving- 
picture industries having grown by leaps and bounds, which 
when analyzed in this way become surprisingly regular and 
uniform. The operation of the law of organic growth in bus- 
iness and economic affairs is even more rigid in national and 
world-wide records. 


3 There is another curve which sometimes fits economic data better than the log- 
arithmic curve, namely the Gompertz curve. ‘This is dealt with more fully later on, 
in the chapter on Special Projections. 


CHAPTER XXXIII 
RATE-OF-CHANGE ANALYSIS 


To subject a business record or the events in any other 
phenomenon to analysis according to the law of organic 
growth is not really a difficult problem. It implies of course 
a comparison of the rate of change or ratios between each two 
successive items in a series, and these successive ratios are the 
successive quotients obtained by dividing each item by the 
item immediately preceding it. This method of successive 
divisions would, however, be tedious even for the shortest 
series, and when applied to the wholesale analysis of a large 
body of statistics, such as (in the individual business house) the 
records of various lines and articles, or (in economics) the 
statistics for many industries or sociological developments, 
would indeed mount up to a forbidding and costly task. Mere 
inspection of the data, from which we could at once detect an 
arithmetical progression or recognize the failure of a given 
series of figures to conform to an arithmetical progression, will 
not often suffice to dig out the geometrical progression. 
Indeed, a fairly close approximation to a geometrical series 
may be so completely veiled in the figures that it passes un- 
noticed through close and even expert inspection of the data. 
The question, therefore, is can the geometrical series be made 
as apparent as the arithmetical one? And how can the failure 
or deviation of a series of data from a straight geometrical 
progression be as easily measured as its deviation from an 
arithmetical one? 

Between the two types of progressions, arithmetical and 
geometrical, there is a curious inter-relation which it is well 
to master. Let us examine again the series 1, 2, 4, 8, 16, 32, 
64, . . . With the exception of the first item in this series, 
it is evident that all the items are merely powers of 2 and 
since, as you know, 1 is merely the 0 power of any number, 
we may include it in the series calling it the 0 power of 2. 

382 


RATE-OF-CHANGE ANALYSIS 383 


Therefore we can rewrite this series as follows: 2°, 21, 32, 23, 24, 
25, 28. . . . Now examine these exponents and you will find 
that they form an arithmetical progression, 0, 1, 2, 3, 4, 5, 6. 

. . . In short, we find that if a geometrical series be rewritten 
as a series of the powers of a single quantity (the constant 
multiplier or rate of change), the exponents of these powers 
will form an arithmetical progression. This is a rule of general 
application and might be used as a means of defining the 
geometrical series. 

Here we come again to logarithms. For a logarithm, you 
will remember, is merely the exponent by which a common or 
universal base figure can be raised to a given value. In re- 
writing the series 1, 2, 4, 8, 16, 32, 64, ... as the series 
2°, 21, 22, 28, 24, 25, 26, . . . we are obviously using 2 as a 
common or universal base for the series and the exponents, 
0, 1, 2, 3, 4, 5, 6, which will raise this base figure 2, to the 
value of the items in the original series may be called the 
logarithms (to base 2) of the items in the original series. 
Since 1 =2°, it is obvious that 0=log,l, since 2 =21, clearly 
1 =log,2, and since 64 =2°, 6 =log,64. In short, in turning the 
geometrical series into a series of the successive powers of a 
common base figure, we find ourselves writing as exponents 
of these powers, the logarithms of the original series (to the 
base figure which was used as the root of these powers). And 
therefore we may say that the logarithms of the items in a 
geometrical series will form in themselves an arithmetical pro- 
gression. This is indeed a very usual definition of the geo-_ 
metrical progression, namely that it is composed of items whose 
logarithms form an arithmetical progression. 

Logarithms can be taken to any common base figure we 
desire. For example, the same geometrical progression, 1, 2, 
4, 8, 16, 32, 64 . . . may be, if we wish, written as a series of 
the powers of 4, as follows: 4°, 4'/2, 41, 4°/s, 42, 45/2 43, . . . Or 
that. same series can be written as a series of the powers of 
any other number provided we but care to do the necessary 
calculating. Now as you know, logarithms are ordinarily 
taken to the base figure 10 and not to the base figure 2 or 4 as 
in the above examples. The reason for this is that our numbers 
are arranged upon a decimal system and by taking the base 
figure as 10 we are able to make the integral part of the logar- 
ithm (characteristic) a mere record of the position of the 
decimal point in the original number, and we are able to make 


384 CHARTS AND GRAPHS © 


the fractional part of the logarithm (mantissa) the same for 
all similar succession of similar digits. Suppose therefore that 
we adopt 10 as the quantity whose powers we wish to sub- 
stitute for the original series. In this case we rewrite the 
series, 1, 2, 4, 8, 16, 32, 64 ... as the series 10°, 10°°, 
1 (P26028 1 99-208 11.120 Nt Again wvettindetheseiexpos 
nents or logarithms, 0.0000, 0.3010, 0.6020, 0.9030, 1.2040, 
1.5050 . . . , forming an arithmetical series. 

The inter-relation between the arithmetical and geometrical 
series is therefore such that in a sense, both series may be 
spoken of as arithmetical, the former being arithmetical in its 
original form and the latter becoming arithmetical when its 
logs are used. For this reason the term logarithmic is often 
used synonymously with the term geometrical to distinguish 
the latter form of progression and its item-to-item changes, 
from the progression which is truly arithmetical in its original 
form. For though the two types of progression are made 
similar by the substitution of logarithms for one of them, yet 
it must be remembered that they are two radically different 
things which must never be confused with each other. In 
their original form or natural numbers the one progresses by 
addition or subtraction and the other by multiplication or 
division and there is a world of difference between them. That 
they behave similarly when logarithms are substituted for one 
of them, is chiefly due to the peculiar qualities of logarithms, 
that by their use the process of addition or subtraction may 
be substituted for the process of multiplication or division 
(instead of dividing one number into another number to get 
a quotient, you subtract the logarithm of the first from the 
logarithm of the second and the difference is the logarithm 
of the quotient). 

In our analysis of our statistics in the light of the law of 
organic growth, we can find a short cut, therefore, through 
the use of logarithms. In other words we can turn the items 
in our data into their corresponding logarithms (consulting 
for this purpose a table of common logarithms) and then, by 
comparing the logarithms, we can quickly discover any uni- 
formity or constancy in their amount of change, and so easily 
detect and measure the degree of adherence in the original data 
to the geometrical series. The deviation of the series of logar- 
ithms from an arithmetical progression with uniform amount 
of change, is a measure of the deviation of the original series 


RATE-OF-CHANGE ‘ANALYSIS 385 


of data from a geometrical progression with a uniform rate of 
change. In practice, this method is very simple and it may 
be regarded as a distinct labor-saving device in the careful 
analysis of statistics. 

In the chapter on Index Numbers, you will recall reading 
that relative figures (that is, percentages) can be substituted 
for an absolute series or series of original data. While these 
relatives are ordinarily computed with a single item in the 
series as the norm or 100%, yet they can be computed with 
each item taken as a percentage of the item immediately pre- 
ceding it. In this case the series is called a series of 
chain-relatives or chain-percentages. Now it is precisely a 
series of chain-percentages which the method of successive 
divisions already mentioned gives us. But you will find that 
the short-cut method of successive subtraction of logarithms 
does not yield results in precisely the same form. For the 
short-cut method gives us the logarithmic differences and 
these differences are the logarithms of the percentages them- 
selves. If therefore we desire to know the rate of change in 
terms of its percentages of change (and not in terms of the 
logarithms of its percentages of change) we must again consult 
our logarithm tables, if we are using the short-cut method, and 
convert the logarithmic differences back into the percentages 
of change (by substituting for each difference its anti-logar- 
ithm). As a matter of fact, however, the chain-relatives or 
chain-percentages are, except for very popular purposes, not 
ordinarily of sufficient importance to justify this additional 
labor. 

You will observe, however, that we have not yet reduced 
the work of rate-of-change analysis of our statistics to the same ~ 
simple and easy steps as are found in an amount-of-change 
analysis, for the use of logarithm tables and the substitution 
of logarithms for the natural numbers requires, even with great 
proficiency, a considerable amount of time and effort. And in 
the wholesale analysis of a large body of statistics by this 
method, we will not only find the work long and tedious but 
we should also expect to find a large number of errors creeping 
into the work which would be difficult to detect. The short- 
cut method has, it is true, eliminated the more difficult proc- 
esses of division (or multiplication), and in the lack of special 
calculating machines, is a long step in labor saving, but the 
rate-of-change analysis is not yet as simple as an amount-of- 


386 CHARTS AND GRAPHS 


change analysis. In the next chapter this final step will be 
taken and through the simple use of the graphic method com- 
bined with the use of logarithms, the work of substituting logs 
for natural numbers will be eliminated and the rate-of-change 
analysis made as simple as the amount-of-change analysis. 

Organic, percentage, geometric or logarithmic change 
(whichever name you prefer) is growth in which the rate or 
ratio of change 1 is uniform. Increment, difference or arithmetic 
change is growth in which the amount or quantity of change 
is uniform. The former naturally forms the basis of judgment 
for the fluctuations of phenomena which cannot be negative, 
that is, which must always be positive. The latter is fre- 
quently the better basis of judgment for the fluctuations of 
phenomena which can be zero and negative as well as positive. 
In general it is perhaps best to study your data from both 
points of view.! 


1 Speaking of the amount-of-change curve, Professor Marshall says: “Its ‘defects 
are such that many statisticians seldom use it except for the purpose of popular 
exposition, and for this purpose, I must confess, it has great dangers.”—Alfred 
Marshall, On the Graphic Method of Statistics, Jubilee Volume of the Royal Statistical 
Society, June 22-24, 1885, pp. 251-260. 


Cuarpter XXXIV 
RATE-OF-CHANGE SCALES 


The rate-of-change curve chart affords in some respects the 
most powerful analysis known of statistical data. An attempt 
has been made in the last chapter to explain the general theory 
of this chart method, but a real insight into its various uses 
can only be obtained from a study of its applications. The 
method is really nothing more than the charting of the logar: 
ithms of numbers in the place of charting the numbers them- 
selves. A careful reading of the last chapter will doubtless 
have already suggested this process to the student, and it only 
remains to set forth the technique of charting logarithms. 
Indeed, as will be seen, such simplified methods have been 
developed that it is not necessary for one to understand logar- 
ithms or be proficient in their use in order to benefit from this 
chart. In the present chapter the development and construc- 
tion of these simplified charts will accordingly be discussed with 
the general principles covering the use of logarithms in the 
charts. 

Three methods are open to us in the plotting of logarithmic 
curves. The first is the obvious one of substituting for the 
items in a series to be plotted, the logarithms of those items. 
We must consult a table of logarithms and for each item find 
the logarithm and tabulate these logarithms in a column beside 
our original series of data. Then on the plain co-ordinate 
paper used for amount-of-change curves, in which the scales 
are arithmetically projected (that is, the scale-figures at equal 
distances form an arithmetical series), we must plot these 
logarithms, and draw the curve through these plotted points. 
The result, of course, will be a curve of the logarithms of our 
original series, or, as we have called it, a logarithmic (or 
“rate-of-change’’) curve. This curve will behave as a logar- 
ithmic curve should and will tell us what we wish to learn 
from the use of logarithms, The straight line, which always 


387 


388 CHARTS AND GRAPHS 


PRICE OF POTATOES 
Average Retail Price per Pound 
United States 
1913-1920 
(Source:= Bureau of Labor Statistics) 


st wn a o 2) 
o Logar- 3 BH 3 a ‘2 1Q 2 = 
Ce chim wee re a <i re) Ye} tas é 
i ° ° e e e e ° e 
i) 
Os ; 
cor} oO ie) 
2 See RE eee 4 
& Number 4 i d ony + ne Sy o 
a 
ae sisal aria (ual 
i ekeael stall 
i BG bevel 


e 
~ 
v, 


a 
Bae 
ENS 


> 

os eee ee 

asia eee at ee ed 
st 


0.0 
te) Ko) é o Q ° 
S qt = ral co a rad Q 
@ ror) cox) con) @ ror) a 
a a a a a a a a 


Fig. 329. The Rate-of-change Curve—First Method. 
This scale carries the logarithms, not the numbers 


represents equal amounts of change and therefore depicts an 
arithmetical progression, will here indicate an arithmetical 
series of logarithmic values and. hence a geometric series in 
the original data—thereby instantly betraying to us the fact 
that our phenomenon has for the length of the straight line 
followed the law of organic growth. And the failure of our 
curve to maintain a straight line will indicate the failure of 
our phenomenon to follow the law of organic growth. All this 
is as it should be, but the method of charting is tedious. 

The other two methods open to us achieve precisely the 
same resulting curve on the chart, but obviate the need of 
turning our original figures into logarithms. No need to 
bother with a table of logarithms, nor indeed, to understand 
the so-called intricacies of such a table. The trick is turned by 


RATE-OF-CHANGE SCALES 389 


merely converting the scale of the chart, once and for all, be- 
forehand, into a logarithmic scale. That is to say, We must 
calibrate the scale figures for the natural numbers, but enter 
these calibrations or scale figures at points on the scale which 
are plotted, graduated, or measured, at the values not of the 
natural numbers themselves, but of their logarithms. Such a 
scale we shall throughout the remainder of this book call a 
logarithmically projected scale. 


PRICE OF POTATOES 
Average Retail Price per Pound 
United States 
1913-1920 
(Source:- Bureau of Labor Statistios) 


Number of 
Cents per 
Pound 
1.7 
1.8 
1.5 
2.7 
3.2 
3.8 
6.35 


Fig. 330. The Rate-of-change Curve—Second Method. 
This scale carries the numbers, not the logarithms, but is not handy. 


_ The first of these two simpler methods uses the same 
arithmetically ruled co-ordinates which we have used for 
amount-of-change curves.! The scale is therefore somewhat 
unhandy. For if every equal interval or distance up the paper 
or scale is to stand for an equal logarithmic value, it must 
stand for an equal arithmetic or natural number ratio. If we 


1For a full description of this method, see Irving Fisher, The Ratio Chart for 
Plotting Statistics, American Statistical Association Quarterly, June, 1917, p. 578. 


390 CHARTS AND GRAPHS 


calibrate the first (ie. lowest) abscissa (horizontal line) as 

unity or 1.0, and let each distance or interval between the hori- 
zontal lines stand for a 10% increase (that is ratio of #3), then 
obviously we must calibrate the second horizontal as 1.1, the 
third as 1.21 (that is % of 1.1), the fourth as 1.331 (that is 
U of 1.21), the fifth is 1.474, the sixth as 1.622, and so on. 
This is what we would call an unhandy scale. It is difficult 
to plot points on such a scale. Nevertheless, it can be done, 
and the resulting curve will be the same as secured by the 
previous method of plotting the logarithms of our series. And 
we have avoided the task of turning each figure of the series 
individually into a logarithm. And by either method you will 
notice that we have been free to make our curve fluctuations 
as high or as low as we wished, by merely selecting our scale 
on a larger or smaller unit length. 

The third method is, however, the best of all, for it makes 
the plotting of logarithmic curves as simple and easy as the 
plotting of arithmetical ones. It consists in using specially 
ruled paper, provided by many publishers of chart paper, in 
which the co-ordinates are unevenly spaced so as to correspond 
with the logarithmic values of the round numbers in the 
original series. Thus instead of an abscissa or ordinate at the 
value of 1.21 (equidistant with the abscissa or ordinate of 1.0 
from the abscissa or ordinate of 1.1), this paper has the ab- 
scissa or ordinate of 1.2, slightly closer to that of 1.1. Like- 
wise instead of an abscissa or ordinate for 1.331 (at another 
equal distance), this paper has the abscissa or ordinate of 1.3 
still closer to that of 1.2. So it goes throughout the scale. 
The paper has been carefully ruled up with these gradually 
diminishing distances or intervals between ordinates accu- 
rately measured to correspond with the true logarithmic dis- 
tances of the round numbers from 1 to 10 and all fractions 
between these round numbers. 

And since, as you have seen, the logarithms of every similar 
succession of significant digits are the same (in mantissa), we 
need merely multiply or divide these round numbers in the 
printed scale of this chart-paper, by any power of ten to make 
the scale suitable for our data. This is the same as saying that 
we can shift the decimal point as far in either direction of 
these printed scale figures as we please, and the paper will still 
be properly ruled off and scaled. Again it is the same as saying 
that we may add or prefix as many ciphers as we want to these 


RATE-OF-CHANGE SCALES 391 


printed scale figures. The changing of the printed scale-figures 
running from 1 to 10 into a scale in which the round figures 


PRICE OF POTATOES 
Average Retail Price per Pound 
United States 
1913-1920 . 
(Source:= Bureau of Labor Statistics) 


Number of 
Cents per 
Pound 


Fig. 331. The Rate-of-change Curve—Third Method. 
The simplified and handy scale of original numbers. 


fit our data is very easy. The only thing to remember is that 
it is done by multiplying or dividihg the printed scale figures 
by a constant—whatever constant we please. In this it differs 
from the changing of scales on the amount-of-change curves— 
in which we could have used addition and subtraction. The 
writing in of ciphers behind or before the printed scale-figures, 
is merely a form of multiplication or division, in which the 
constant is some power of ten. It is indeed perfectly possible 
to use for our constant some figure which is not a power of 
ten. But we must multiply or divide by this constant—we 
cannot add or subtract. 

Now it often happens that a scale running from 1 to 10, 
that is in which the maximum of the scale 1s ten times the 
minimum, does not afford us sufficient range for the fluctua- 
tions of our data. Were we to attempt to plot our curve upon 
this specially prepared paper, we would find that the curve 
would quickly run off the chart. To this problem the answer 
is very simple. We merely join together two sheets of this 


392 


CHARTS AND GRAPHS 


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84% x 11 
k pap 


RATE-OF-CHANGE SCALES 393 
paper, or two sets of these rulings, and recalibrate the upper 
one by adding an extra cipher to its scale figures. In this case, 
the entire scale, through two sets of rulings, runs from 1 to 
100—that is, the first runs from 1 to 10 and the second runs 
from 10 to 100. Simple, isn’t it? Not only two, but many 
of these sets of rulings, can be joined together in this way, 
giving us a scale range of from 1 to 1000, 10,000, 100,000, 
1,000,000, or more. In fact, the publishers of chart paper 
have anticipated this need, and provide paper with these sets of 
rulings combined two, three, four and sometimes more, upon 
a single sheet. Each set of rulings is called a “deck.” The 
single-deck paper runs from 1 to 10, the double-deck from 1 
to 100, and so on, two and three-deck papers being the most 
generally useful ones. 

These considerations of the number of decks needed for a 
chart are all based upon the range of fluctuation in the series 
to be plotted. Before determining upon the number of decks 
to use in your chart, you must first glance through your series 
and note not merely its highest but also its lowest items. If 
both have the same number of (integral) digits, a single deck 
is sufhcient; but if the maximum has more (integral) digits 
than the minimum, you need as many additional decks as there 
are additional digits in the maximum figure. A little thought 
will enable you to determine exactly the number of decks you 
need before you start your chart. | 

A very different problem is the size of decks used in your 
chart. In order to make their chart-forms of uniform over-all 
size, the publishers of this paper are accustomed to make the 
decks smaller as they join more of them together. Thus if in 
single-deck paper the chart measures six inches to the deck, in 
double-deck paper the deck will measure three inches that the 
chart may still measure six, and triple-deck paper will have 
three two-inch decks and in four-deck paper the four decks will 
each measure one and one-half inches. This, for general ap- 
pearance, is excellent, but you must not mix the various sizes 
of decks in the same report or set of charts. You must main- 
tain a uniform size of deck, for a chart upon a three-inch deck 
cannot be compared with one upon a two-inch deck. The deck 
on the smaller scale will show the same fluctuations smaller 
than they would be upon a deck on a larger scale. So if you 
use, let us say, the two deck paper in which each deck meas- 
ures three inches, you must keep on using that paper so long 


394 CHARTS AND GRAPHS 


as you are charting statistics to be compared with it, regard- 
less of whether your data at times calls for only one deck or 
for three or more decks (in this latter case you must make up 
three or more deck paper to fit the two deck paper, the decks 
being uniform and the chart larger). So it is best, therefore, 


400 == 


a7 ===> >> —— 

0 SS 

326 —_——_=$ = SS SS SZ 

—— 

Aa ==>======= = = 

—— ————— Sel SSS Se 

de | ong ao oem aeas ease tiee Soe Soe Sos ae == SSS SS = 

 ———————— —— —— — —— — —— ——————— —  —————— ——— 
iia eS ee PSS) eo ae os 9 | aes sa [oes EE ee ie 
th 

200 p——}—_f ff NT ee Er Se I ee 
ES Ses ey SS RS SS Se) SS SS ee ens see ee a ee 

175 (Sar) eee ae Bee) | Pe as Se SS ed ee }-—f f+ —_f f+ 
hen PRS | Be ae a Eo! ea eee BAS 
ee 
(2a) DS ae ES ea Eee =e 

150 ee ee ee ee se ff —_} —_}|__+}__ 
eS FSS PS ee oe ee a t+—] ata a] Rs Ds a 
iets ee Gd RCo Cond as nea Ee (eel iad fi es SS EE ee 
a ie lee == a) ised i al ee ae ee es 
aed BE a ES a (ea ed Se De OO eh el es (el ee el ee ee SS eS 

[esas Sse (aaa eal aT (mal SS a PPT (EN a Ed 
te FRE fe SEC (es ee en Pe re 
jae Se a Em acl ea ed Se len el ae ae ed Be 
SE (ae a a I a Se) ee ee ee ae 

i i a a el Ae RL Rk ADE A Le 
SR a LM a a (ee a a ee ee A A eS 

80 oD + re) o ~ ioe} ao Land a ~ = Ld snl a o bs al 2 
- co Coal d o o oo o o 


1930 
1936 


o 
=~ 


Fig. 333. A Rate-of-Change Part-Deck Form. 


This form, designed by the author, for price-fluctuations, is used by the Bureau 
of Labor Statistics both as an office-form and in its publications on prices. The 
horizontal faint-rulings (those without scale-numbers) are blue in the original, 
so as to disappear in reduced reproductions. 


before preparing a series of charts, to inspect all your data and 
pick once for all the best paper suitable for the most widely 
fluctuating series in the data. 

In addition to the one, two, and three or more deck papers, 
you may occasionally need, and can also obtain from some pub- 
lishers part-deck paper. ‘This is useful for showing fluctua- 
tions which do not cover a range of more than 1 to 3 or 4, that 
is, in which the maximum item is not more than 200 or 300 per 
cent greater than the minimum item. Here you may find that 
a whole deck would waste paper and fail to show the fluctua-. 
tions clearly enough. You may therefore feel called upon to 
adopt a chart-ruling covering only a part of a deck. But, as 


RATE-OF-CHANGE SCALES 395 


will be later shown, it does not pay ordinarily to carry this 
detail too far, for as you take a smaller and smaller part of the 
deck you will find the rulings approaching nearer and nearer 
to plain arithmetical or uniform distances and your curve re- 
sembles more and more closely a plain amount-of-change curve. 


WETALS AND METAL PRODUCTS: Copper, Ingot, Electrolytic, New York, Monthly, 1913-1918 
(Before 1007, Lake) 


ANnONnNmRenon 

Rie tetas ce ioe gadsdaddedeedencanedcedceesass 
oar NNRDs Aa mOnd 8esss 

ice FERSSSESSS ESS SS SSS SEES SRES SA Raa AAAS SEGRE SSESES 
ANFAMONMNMDOHODNDO ‘a Sone 

tee ger GUS ase saau assess Ss Qaceesegsgeggesersssagsssesss 
price cer SATA AAAS SAA ASA ARAN AANA AAAS EA REREAN RR RSRRSSR SS 
eee eee ee eee eee eee oe eee eee ee oeseeeee ee eee 


400 
375 


350 See SS=5 52525 
$25 — = 
BESE5= = a oe S255 
300 Bas=========== = es —— 
Wig S=SSesnes=s=S==== SS5SSSe=5ss=0=====25==552 
$e s=sasass==es=aa== = Sane sa==sa-==S== 
250 == = === 
SS 22522552 Bee = S=S=Ss= = 
925 | = 
=e 
Ee SeESarreeeerseseze 
ve FEES SESE Senssceessarsee2 
Paasche cesaae Seeeee Seen GS man ie 
nese = 4 i ea sabe 
450 = 4 sans eaeas EEE = wa 
SSEESEcostdt cesteactitoe i tossttons 
as> 5 PEEEEEEEH H BG soeeo 
3 PEE EEEEEEEEEEEEEEEEEE 
aa BERSEe BEER Eooo sees 
GEER RS mg Bee Senn) ie 
EAGEEEeRP Zee bem COOP 
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‘> BEDE GEE Ceo Bis NET Tbe TT 


Adds sslargsissga sala Tears 


fgds2is2ggisgaddazbagksia 
1913 1914 5 


1916 


Fig. 334. Part-Deck Rate-of-Change Paper. 


The curve-chart record form used in the Bureau of Labor Statistics for price 
relatives. This is an office form—in this case filled out for electrolytic ingot 
copper prices at New York—the chart nearly filling 84 x 11 inch paper sheets. 


There is, however, still another form of ruling which you 
will often find useful, known as the “split-deck.” By means 
of a split-deck you can often keep to larger decks though 
your maximum and minimum figures in a series lie in addi- 
tional decks. The split-deck is merely the upper half of one 
deck and the lower half of another joined together in a single 
chart. You may have split one-deck paper or split two-deck 


396 


100% 


(Sourcee:, Korthly Labor Review and U. S. Stat{etioal abstract) 


United States, 1640-2920. 
1912 average 


WAGES, PRICES AND MOBEY IN CIRCULATION 
Relstive figures of hourly wage-rates, retsil food prices and percapite money in cirovletion 


250 
aoc 


circulst. 


First Quarter 


Last Quarter 


A “‘Split-Deck’’ from 30 to 300. 


gle deck to 20-200 or 50-500. 


This 30-300 scale was specially drawn, from a slide-rule. 


Fig. 335. 


1S easier to convert a sin 


It 


RATE-OF-CHANGE SCALES 307 


paper, or more, but the first is the only kind ordinarily pub- 
lished. By means of a split-deck chart-field you can often 
keep to a larger size of deck and yet show a curve whose 
maxima and minima lie in different decks. Any paper can be 
converted into split-deck paper merely by multiplying the 
‘printed scale figures by some constant other than ten or a 
power of ten—two and five being the usual and most con- 
venient constants for this purpose. 


If you do not have access to the marketed forms of spe- 
cially ruled logarithmic chart-paper, you can readily prepare 
it for yourself, either by plotting for your scales along the 
axes, the logarithms of the round numbers, as found by con- 
sulting a table of logarithms, or by copying the calibrations and 
graduations from an ordinary slide-rule. If the slide-rule (in 
which the deck is usually about five or ten inches long) does 
not have the right size of deck to suit you, and in general if 
you wish to alter a scale as to size of decks, you can accomplish 
this by the trick of laying off (or “‘projecting’’) the given cali- 
brations to precisely the size you desire by the use of parallel 
lines from the given scale to the desired scale, across a triangle 
formed by the two scales and the last parallel? 


It is a pretty way of ornamenting the page on which a 
rate-of-change chart is shown, to mark off a short additional 
scale near, but not as part of, the chart. This scale need not 
be as long as the scale upon the chart. It may be made up of 
two parallel lines close together, with cross-lines at the round 
numbers on the scale, the whole looking very like a long narrow 
ladder. If you wish to make it more conspicuous, the alter- 
nate spaces between cross-lines can be blacked in. The virtue 
of this gratuitous and emphasized scale is that it calls the 
layman’s attention to the strange and unusual (to him) nature 
of the scale of the chart. And its strong markings also show 
the constant significance of distances upon the chart, regard- 
less of height. By recalibrating this extra scale with scale- 
numbers of “per cent increase or decrease”’ (that is,O at the 
point of 100, 50 at the point of 150,-25 at the point of 75) you 
make this constant significance of distances even clearer. The 
small extra scale is then an excellent device for use with a pair 
of dividers—it is like a scale of miles on a map. The reader 


2 Cf. Figs. 166 and 167 on p. 185. 


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geeceeseeeeaes SS G1 S.8 > 85. 8 2 Sa ae a a Se ee 
oe (2047 8qN TPOTISTIMIS “S “N =:30sN0S) 
ON Boziewy Jo Aisacostp eq} SOUS PT tom 993 UT pTO3 Jo aofzzonpoid peqemyzeq 


8) NOILondowd 100 $,@1HOK SAL 


Ye 


RATE-OF-CHANGE SCALES 399 


can adjust his dividers for any two points, hold the dividers 
to the scale, and read the percentage relation between them. 
And this, after all, is the purpose of rate-of-change charts, 


Nore to Fic. 336 


Fig. 336 illustrates the computing possibilities of rate-of-change paper. The 
data, as will be seen above the chart, is for periods of irregular length as shown 
by the dates below. While the points have been plotted for the data, these points 
cannot all be connected as a single curve. It is necessary, as pointed out in the 
chapter on frequency curves, to find averages for periods of uniform length. 
We need not, however, calculate these averages—on the contrary with a pair of 
calipers or dividers we lay off the proper distances below the points from the special 
scale at the right (or, in its absence, from the regular scale at the left) and obtain 
at once the plotting points for a single connected curve. In this chart two such 
calculated curves have been drawn—one for an annual average and the other, 
just one deck higher, for a decennial ayerage. 


CHARTS 


we 


i \ < - 7 
a ee ee Sef a iaenas elahain snail Om aimlenleaeais sinlmini ol 


keto ; 7 ke--W76---.-->t 


10 

2 
8 
7 

5 6 
5 

4 


Fig. 


RATE-OF-CHANGE SCALES 401 


337. One Way to Find the Rate-of-change Scale. 


On any horizontal distance OX lay off OA equal to one-tenth of OX and project 
a semi-circle on OX (see upper left-hand diagram); drop a perpendicular 4b from 
OX to intersect the semi-circle at b. Then 04:0b::0b:OX and 0b?= (OA) 
(OX) = (1) (10) and Ob—= V/10. Lay off OB on OX equal to Ob (see upper 
right hand diagram) and drop a perpendicular Be from OX to intersect the semi- 
circle atc. Then OB:Oc::Oc:OX and Oc?= (OB) (OX)=10 V10 and Oc= 


VW 10V/10. Likewise (see third diagram) lay off OC, OD, OE, and OF on 
OX. These distances (see fourth diagram) represent V10 v/10, 1 10V/ 10 V/ 10, 


10V 10/10/10 and 10 10V 10/10/10 respectively, or 10%, 
10%, 10%, and 10%%2, while we already have in OA, OB, and OX, obviously 10°, 
10%, and 10!. Similarly (see fifth and sixth diagrams) we can find 104, and 
10%, 10%, 10742 from semicircles upon OB and OC. The ordinates from these 
points (see lower left-hand diagram) intersect abscissae from a scale of the expo- 
nents, 0, 14, 34, %, etc. so as to form a logarithmic curve, or curve of the loga- 
rithms of numbers, in which y==log x. By taking abscissae from the intersect 
points of this curve with ordinates from the even numbers from 0—10, we find 
the logarithmic scale of these numbers (see lower right-hand diagram). 


CHAPTER XXXV 
RATE-OF-CHANGE CURVES 


From the construction of the logarithmic chart, we turn 
naturally to the general principles of its application, and here 
we must consider two of its limitations. The first of these has 
already been touched upon, that the utility of the logarithmic 
projection is limited to data in which the range of fluctuation 
is fairly great. There is not much to be gained from the logar- 
ithmic projection when the maximum of a series does not vary 
by more than a 100°% from the minimum of the series. For the 
logarithmic scale and the arithmetic scale approach each other 
more and more closely as the percentage between the limits is 
made smaller and smaller. Within a range of 100%, that is, 
when the maximum or upper limit is not more than twice as 
great as the minimum or lower range, the difference between 
the two projections or scales is not great enough to justify 
ordinarily the effort of the less usual method. Within a range 
of 50% variation the approximation of the two projections or 
scales is very close indeed, and within a range of 25% varia- 
tion, the difference is hardly noticeable. The real value of 
the logarithmic projection is for data which fluctuates to ranges 
exceeding two or three hundred per cent or more. 

This fact, that when only slight changes take place in the 
total series under observation the geometric progression closely 
approximates the arithmetical one, enables us to use the 
latter, because of its simplicity, even for the purposes of inter- 
polation and extrapolation. Thus while the population of the 
United States increases by about two per cent per annum, a 
geometrical progression, yet the Census Bureau employs con- 
stant differences or amounts of change in estimating the local 
population for the inter-censal years. The results are sufh- 
ciently reliable, because at two per cent it would take the geo- 
metrical progression many years to outdistance the arith- 
metical progression noticeably. And in the charts which fol- 

402 


RATE-OF-CHANGE CURVES 403 


low, designed to show geometric progressions instead of arith- 
metical ones, it must be remembered that no great benefit is 
secured from charts in which the range of fluctuation of curves 
is slight. 
ANNUAL 
RATE 
100 - ————— = — — 
90 
BO 
70 


60 


50 


Vv 
» 


40 


eel 
_k 


30 


20 


1919 1920 1921. 1922 


Fig. 338. Comparison of Series Lying in Different Parts of Chart, Though 
Not Fluctuating Greatly. 

Annual rate of turnover of bank deposits in representative groups of banks in 

different cities —Permission of Mr. Carl Snyder. 


To this limitation of the usefulness of the logarithmic 
chart method, we must note an important exception, arising 
when a number of different series are to be compared and al- 
though their individual fluctuations are slight, yet they would 
lie upon far different portions of the chart. In the chapter 
on Index Numbers, you saw that such quite different series 
could be easily compared by reducing them to percentages, the 
change into percentages or relatives making their fluctuations 
comparable when charted upon arithmetical or ordinary 
amount-of-change chart paper. The use of the logarithmically- 
projected chart scale, that is, the rate-of-change paper, will 
however obviate the need for mathematical computing re- 
quired to change the series into relative percentages, as the 
logarithmic projection makes their fluctuations comparable 
regardless of their position upon the field of the chart. Thus 


404 CHARTS AND GRAPHS 


FARM AND FACTORY WAGES ; saiphh e hed 
farm labor wages (where board was not included) in the Un ates 
pilav with Srokepa veckly earnings in representative factories, N.Y.state 
1910-20 
(Sources:* U. 8. Depts of Agriculture and N. Y. State Dept. of Labor) 


” ° o rr) 
Farm Labor ° = 8 a & cI 3 < % “ a 
by the month A o a oO a o ry oO o oO ~ 
” 7 < wo © 

without board ‘% nN OY bY Oy bo 
~* \ wo ° w 
Pactory workers < © ° 3 8 8 a 
(office and shop) a 33 < 2 3 9 2 

weekly earnings 

Bete tast easton 2 ie ie a s x 
Fare labor ty ee be or ei ee eee ho MO Oe eee 
day (not harvest) 5 a a a a a a a a "9 oy 


without board 


we we eee eee 


50 


40 


30 


20 


10 


1910 
11 
12 
13 
14 
15 
16 

7 
8 
19 


Fig. 339. Amount-of-change Curve (Absolute). 


two series of data which fluctuate similarly as to rate of change, 
appear very unlike when plotted arithmetically, if one lies 
further from the zero or base line than the other (as when the 
units of measurement of one are millions of dollars, and of the 


RATE-OF-CHANGE CURVES 405 


FARM AND FACTORY WAGES 
Average farm lebor wages (where board was not included) in the United States 
compared with average weekly earnings in representative factories, N. Y. state 
1910-20 
(Scurces;e U. S. Dept. of Agriculture and N. Y. Stute Dept. of Labor) 


~~ - ~~ ~ -~ -o- 
Farm labor ° S e = @ - ey fe -S = = 
Qe oOo OH ww Pt Lr} ° fo wo 
by the month TE OIE BES Cs es Rts Eee a> Rae Cot en an 
without board 3 Sy a CU Sa in Oe a Gia! | 2 mae Olr anasic2 
Factory workerna o oe ° S ey oO is 
(office and shop) <8 Rew SO BO Seay) Te ics 
weekly earnings a miles a SS es DS 
wines ws taisie ws aieloand — ~ — ~— — — — 
~7_- ~~ 
Farm lebtor by the 2 HS ie Sy oe cr) SS im 2 S a ion 
A) a 
day (not hervest) 32 3 “O “2S <8 Sg Shes. Shea RR Giller, Cee 
without board - - - - L oS Oe OS 
ee ee ew we ee ee ar 
| | | a ral [ 
250 : | | 
Soe od eae ae 
vA 
ai ce 
a Seti = z 
shea ace = ae: 
200 aie ak 
—— 


LIAS 
mS 
nN 


: ai 
tt 
Re 
4 
| 


1 
[ 
ANE 

oe 


100 |—_— --->~-~ BS ee ; | S| 5 
ee ZS 
F = 


inate iad 
cous ws ee | [ewer : eeu 
50 (as a 


o Lod tue) Qn 
Leal nr « = 


11 
1 


Oo 
a 
@D 
- 


Fig. 340. Amount-of-change Curve (Relative Numbers). 


1320 


other dollars and cents). But when logarithmic paper is used, 
these two series appear to fluctuate together (just as they did 
when reduced to index numbers or relatives). The logarithmic 
chart method moreover avoids all confusion which might arise 
as to the base year or period employed for the two relative 
series. This advantage becomes important when different 


406 CHARTS AND GRAPHS 


7 FARM AND FACTORY WAGES 
Average Ferm labor weges (where board was not included) in the United States 
compared with average weekly earnings in representative N.Y.state factories. 
j 1910-20. 
(Sources:- U. 8 Dept. of Agriculture and N. Y. State Dept. of Lebor reports) 


Websteber Ss Sl eg ea ee eee 
by the month e © a fo) o ° oa 3 o o + 
without board a nu nu "9 ) ” bey + + wo o 
Factory workers 2 8 3 DS 2 9 2 

(office and shop) Be & x 5 a - #) 
weekly earnings a a a a a a x) 
Here Javon ey ee Oo eS. Sk St eee ee 
day (not harvest) ° Ee ae fe x & a ° x a Ke 
Lied Sobek aS aie eS ee ee eee 
arty a, Bl [Sa a a [RN [SR RS Bl in esis ie, ee |, 
80 
Es Td eed SN es A rs Ses Ee 
es ESS7e AO | ee ons eek ae LR) EO GS A 
, APO see i es SE 
ee eae —~ aes pee a |S 
et. - 
ESS : i 
cA raed eRe ee eee eee : 
6: ( eae [a ole 
ees Paeted ela a Ss 
6 Eieds Dalen Mes (Perc | ee Cal 
Tak Ee? elie ee ae oS, 
‘ ee Bor ae 
lee 
Lae ae Fe 
2 ate 
is eee Pla 
1 
g a x 3 = a a 7 ee 
2 nH 
La, ot 


Fig. 341. Rate-of-change Curve (Absolute). 


periods of time are used as the bases for the two relative series, 
for even identical series would differ from each other with dif- 
ferent base figures when plotted upon amount-of-change paper. 

The other limitation of the logarithmic or rate-of-change 
chart (and a much more important limitation) is that this 


aon 
RATE-OF-CHANGE CURVES 407 


chart can be used only for values in which zero is an absolute 
limit. The chart cannot be used for data in which the values 
cross from positive into negative numbers or vice versa. Zero 
is an absolute limit to any geometric progression and to any- 
thing operating under the law of organic growth. The popu- 
lation of a community can never be zero (if it is to remain a 
population) and it cannot be a negative quantity. The pro- 
duction of a factory might be zero, but it cannot be a nega- 
tive quantity. The sales of a concern cannot be negative. 
Innumerable examples could be given of data to which zero’ 
is an absolute limit and to all such data the rate-of-change 
chart is not only applicable but proper. 

To the rule that zero values cannot be shown upon a logar- 
ithmic chart, there is an apparent exception, to be found in 
the case of data measured in units which are made upon an 
arbitrary and not upon an actual, zero-point. The Fahrenheit 
scale of temperature, for example, places its “zero” value a 
short distance below the freezing point for water at sea level. 
Zero here is an entirely arbitrary valuation and does not mean 
an actual nothing, that is, it does not mean zero heat. To plot 
temperature by taking the logarithm of Fahrenheit degrees 
themselves would be ridiculous. Not only would we be unable 
to show zero degrees Fahrenheit (because the logarithmic scale 
cannot reach zero) but indeed the shape of any curve which we 
might plot in this way would be meaningless. The sensible 
procedure would be to plot the temperature after changing 
the readings into the absolute scale of temperature, or more 
simply, to prepare a special scale in which the degrees were 
plotted at their values on the absolute scale. This is done by 
graduating the scale according to the logarithms of the ab- 
solute degrees of temperature, and then recalibrating or label- 
ling these graduations with the equivalent Fahrenheit read- 
ings. After this recalibration or special labelling the figure O 
would of course appear upon the scale of the logarithmic chart, 
having been entered at the scale-point which really represented 
about 265, the point on the absolute scale corresponding to 
O° F. This example makes clear that a zero reading or value 
can appear upon a logarithmic chart when it is fictitious and 
really represents a positive value. Another example of the 
same type would be a scale of time which included the year 0 
A. D., from which we date our years in modern history. All 
such are cases in which the real values plotted upon the chart 


408 CHARTS AND GRAPHS 


have actual positive values, which through some peculiar 
circumstances must be assigned zero or negative values to 
conform to ordinary practice. 

A more general example of this recalibration of the scale 
resulting in zero and negative values is to be found in per- 
centage scales in which the hundred per cent point or line has 
been relabelled zero and all other figures on the scale corre- 
spondingly relabelled to represent percentage of increase or 
decrease from this particular point. What has really happened 
here of course is that 100% has been subtracted from every 
point along the scale in order to get the desired calibration. 


900 1000 
700 800 
500 600 
400 500 


300 400 
200 300 
100 200 

3 

> .0 100 

§ -20- 80 

A -40 60 

$5709 50 ng 

© 60 40 § 

oO 

2 70 30 8 

2 + 

§ -80 20 g 

4 

: 

= -90 10 

£ -92 8 

® 

& -94 
-95 $ 
-96 4 
-97 3 
-98 2 
-99 1 


Fig. 342. The Percentage-Increase-or-Decrease Recalibration. 


It is an apparent (though not a real) exception to the rule 
given in an earlier paragraph on the construction of logarithmic 
charts in which scale changes were said to be made only by 
multiplication or division of the scale figures, and not by ad- 
dition or subtraction. Reading upward on this new ““per- 
centage increase or decrease” scale from “0” (entered at the 
true point of 100), we find “+50” entered at the true point of 


| RATE-OF-CHANGE CURVES 409 
150, “+100” at the true point of 200, “+900” entered at the 
true point of +1000%, and so on. Reading downward we 
find ‘‘ —10%” entered at the true point ‘of 90%. “=50%,,” 
entered at the true point of 50%, “ —90%” entered at the 
true point of 10%, and so on, the chart approaching but never 
reaching the point of —100%, which belongs to the real zero 
value which cannot be shown upon the logarithmic projection. 
Such a recalibration as this special “percentage increase or 
Mortality Raves per 1,000 Population Bt apecibled seektent® 


1910-1912 
(Source;- United States Bureau of Census) 


wo 
oe uN OO! UM w ow 
Spee ate eee = Ses Ss 2 eu sre eras 
Falls SO Sie cer CM COMERS bed) ict) cety ceple 09h Us ot cp ep gaa leans a Anse 
* Z A nxn nh oO oO A 
Eee ae} 
: Pe oomess ce te ea es BS ANS SS. Suite 
Drowning Sy ey EIS te ee She ict Ss i) > 
4 ---=+ 2 
co x 
Oo vA Ga &=& + = 
SESE EC ona Geos SU re ae Te 
Railways S96 Oo A ® © 9, # mom 8 e Ba wt oe 2 
Secvesmae 
oe 4 
S Sipe o eos Be 8 88 83 42 8 eae 
3S iy esa Ys) Ce WO 
Falls SE OoOD Ros gOm ONE OF OF LO: Cr ee ae 
4 $ 4 et 
2 Ont Oy Oa Ot Ce 2 
~~ + 9 aD Mac 
2 pope e ts CCS Maresh Se Sars eR Ce Bee ye 
g Drowning Cue OmmOl (On JOP. ST OF a OF FO EOS BOT S 
8 es 
D 
yn xt Ho EG Gee Sok Se: eae 1 8 
gsgrseseS8 882288 8 of 
Railways Soe Onsen ONTO KO) FOS fo 
eateries 
8 
4 
2 
‘i 1, 
+8 
oA 
oe 
al 
208 
+04 
\ 
202 
: 
' 
401 see : 
° S : 


Ysars cf Age 


Fig. 343. 


moe CHARTS AND GRAPHS 


decrease” scale, though possible, is really little used and in 
actual practice more or less exceptional. In common with 
the examples given in the previous paragraph, it is always a 
little puzzling because the uniformity of ‘“decks’’ has appar- 
ently been destroyed. 

To the fact that the logarithmic charts cannot show a true 
zero point, we owe one of its most important and unique fea- 
tures, namely that, the height of a curve upon this paper 1s 
entirely without significance and the curve may be moved 


MARRIAGE AND DIVORCE 
Number of marriages and divorces reported and total population 
United States 
1887-1916 
(Source: Census Repert) 


| 888885 88888888888888888 
Population 2338883833383 33888S885383ss3s3 
SoseeeeeresssSsssesenasa 
wring SESSSRR ESSER SRERER s 
SeReSSE SSSIsTsesees3 3 
| 88 
Divoroe Ps" g 


200,000,000 2,000,000 200,00¢ 


150,000,000 1,500,000 160,000 


100,000,000 1,000,000 100,000 


80,000,000 800,000 80,000 


60,000,000 600,000 60,000 


40,000,000 400,000 40,000 | H 


20,000,000 200,000 20,00 


Fig. 344, Several Scales in a Single Split-deck,. 


+2) ee 
i 
. 


bodily up or down upon the chart without altering the sig- 
nificance of the curve-fluctuations. When we say that the 
chart cannot show a zero point, we mean that you could con- 
tinue the deck and ruling of the chart paper downward infin- 
itely far without ever succeeding in reaching zero. You would 
merely reach smaller and smaller fractions of positive values. 
The true zero-point is located out in infinity. It is therefore 
taking no liberties with the chart to slide one curve up or down 
as far as you please to make it more easily comparable with 
another curve, because you are not really changing the position 
of the zero point (that still lies for both curves out in infinity). 
If the two curves are upon separate sheets of paper we may 
slide one piece further down than the other so as to bring the 
curves into close association with each other. Likewise if we 
plot both curves upon the same chart we can use a small scale 
for the lower curve and a much larger scale for the higher 
curve, and so superimpose one upon the other. (In this case 
separate scale figures may be an advantage to the reader but 
they are not essential.) This juxtaposition of curves is one of 
the chief advantages of the rate-of-change paper and can easily 
be carried so far as to make one curve cross or intersect the 
other. It is however considered better practice to prevent the 
crossing of two curves which have been brought together in 
this way, by sliding one curve lower down and altering the 
scale correspondingly. If you are intent upon a very clear ex- 
position of the artificial nature of ‘this juxtaposition, you can 
wipe out a small portion of the co-ordinates of the paper 
between the two curves so as to indicate a break or omitted 
portion of the chart-field between them. 

From this arises a very important use of rate-of-change 
charts in the detection of correlation. For not only is the dis- 
tance or interval between points upon the logarithmically 
projected chart useful for comparing the successive items in 
the same series of data, but it is also useful in comparing 
corresponding points upon different series. ‘The problem here 
is not to scrutinize the slope of parts of one curve, but to 
scrutinize the distance between twocurves. ‘This distance 1s, 
as you remember, when a distinct parallelism or mirroring of 
the two curves is noticeable, an evidence of that similarity of 
behavior which is called correlation. To some extent, corre- 
lation can be discovered by the use of index numbers, or re- 
lative percentages, which make the fluctuation in different 


RATE-OF-CHANGE CURVES _. 4II 


» Students in colleges universities and 
rye6 of various sizes (over 300 volumes each before 
populetion in the United States, 


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1870-1920 
(Source:- U, 8. Statistical Abstract) 


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CHARTS AND GRAPHS 
CULTURAL ORORTH a THE UNITED STATES 


and volumes in libra: 


1890 and over 1000 volumes thereafter) comp 


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To obtain 


of-change curve. 
gree of correlation between var 


But a far more 
sary to fall back upon the mathe- 


parable. 
fforded by the rate- 


an exact measure of the de 


d 


f course neces 


parison is a 


Fig. 345. Shifting Curves to Avoid Insignificant Crossings. 


ata, It is o 


. 


sets of data more com 


ey 
wz j 


matical operations which yield a “correlation coefficient.” But 
fer most purposes the graphic method afforded by rate-of- 
change charts is sufficient and it is always, of course, much 
easier and more rapid. 

When a great number of series are plotted upon rate-of- 
change curves they can be compared in short order merely by 
inspection. If they have been plotted upon paper which is 
sufficiently translucent, closer inspection can be made by 
the use of “light Eee that is by laying one chart over 
the other, holding the two of them up to the light, and sliding 
one back and forth and up and down until it most nearly 
coincides with the other. Mirroring can be detected by turning 
one of the charts upside down and bringing them together for 
light analysis. The great advantage of the use of the rate-of- 
change chart for correlation detection is due to the fact that 
various “lags” in the correlation of the fluctuation can be im- 
mediately corrected by this method, whereas by the long 
mathematical process, a slight lag might be sufficient to wipe 
out and conceal any correlation which might exist, even 
though that correlation be most complete. Even when the 
mathematical processes are to be used, in order to measure 
the correlation exactly, it is best to use the graphic method 
first, in order that the mathematical work may be performed 
only upon the series showing appreciable correlation and in 
order that any lag which may be present can be corrected for. 
In short, for the work of correlation studies, so essential to 
forecasting, the rate-of-change curve chart is becoming recog- 
nized-as necessary. Extrapolation has already been discussed 
as a means of forecasting or predicting the nature of future 
developments and this paper will be found admirably adapted 
to such work, being statistically indicated wherever the fluc- 
tuations of data are logarithmically more regular than they 
are arithmetically. 

A word may be said as to terminology. The curve charts 
which we have previously examined under the general name 
of “amount-of-change charts” are sometimes called increment 
charts or difference charts from the fact that the fluctuation 
of the curve plotted upon them represents increments added to 
or differences subtracted from their previous values. They 
are also sometimes called arithmetical charts, from the fact 
that the straight line upon them represents an arithmetical 
series. The charts to which we are coming and to which we 


RATE-OF-CHANGE CURVES 413 


Pa 


414 CHARTS AND GRAPHS 
oe HE Eee | | ett 
: iit PEPE Cree Cee Cee Eee EEE 
so i. ia a a 
: FEE : EEE EEE 


g 3 3 3 


YEARS. ‘ 
ACTUAL POPULATION OF THE UNITED STATES. DIFFERENCE METHOD. 
Showing the impossibility of correctly comparing rates of increase at different periods. 


1660 


1800 
ao 


RO SR8 Fs 


oo 


From Irving Fisher's ‘The Ratio Chart,” in American Statistical Association Quarterly, June, 1917. 


Fig. 346. An Amount-of-change Chart. 


give the general name of “‘rate-of-change”’ charts are some- 
times called ratio charts! from the fact that the fluctuation of 
the curve plotted upon them represents in a certain sense the 
ratios of change rather than the amounts of change. The 
name is a poor one because the ordinary bar-chart and the 
ordinary amount-of-change curve both express more graph- 
ically the actual ratios between the quantities; the rate-of- 
change chart showing graphically only the changes of ratios 
but not the actual ratios themselves. These charts are most 
frequently called logarithmic charts from the fact that they 
show logarithms and not the anti-logarithms or natural num- 
bers. 

The real distinction between the amount-of-change curve 
and the rate-of-change curve is the distinction between quan- 
titative and qualitative analysis of data. For a quantitative 
analysis, that is a study of the actual quantities involved, the 
amount-of-change paper is necessary. But for a qualitative 
analysis of the figures, that is a study of their comparative re- 
lations, ratios, and proportions, the rate-of-change paper is 
necessary. The fundamental distinction to keep in mind is 
that this qualitative rate-of-change paper illustrates relative 
or proportional changes and is significant only as to them. 
Moreover, because this paper does not illustrate totals, it is 
always well to have at least your important data plotted both 
upon amount-of-change curves and rate-of-change curves, that 
from each type of curve you may easily get its particular sig- 


‘The name, it is believed, was introduced by Professor Irving Fisher, obit. cit. 


00 
«00 
70 


g 3 3 a 3 3 3 3 8 3 
YEARS 
'HE SAME. RATIO METHOD. 
Showing clearly the slight deviations, since 1860, from a uniform rate of growth. 


From Irving Fisher's ‘The Ratio Chart,” in American Statistical Association Quarterly, June, 1917. 
Fig. 347. A Rate-of-change Chart. 


nificance. And it is because the qualitative study, that is the 
study of relative values, is so frequently the more useful one 
that the rate-of-change charts are themselves so commonly 


more valuable.2 


2 The significance of the logarithmic projection need not be difficult to understand’ 
The measurement of star-light in magnitudes, and the measurement of sound-waves 
by means of the octave and its parts, are familiar, examples in which we have adopted 
logarithmic (or exponential) units of measurement, as clearly necessitated by the 
type of the phenomena. It is permissible to think, therefore, that in all cases where 
the logarithmic projection is found suitable, nature is operating in logarithmic units, 
that is, changing organically in geometric progression, while man is still thinking in 
terms of arithmetical units, and must needs project these logarithmically, 


Cae aah ins Ni tales PTR Ce pe oe Se ee eee eV oe eee ee a ee eae Te 
obi : 7 Y } 7 vk a yy rere 


CHAPTER XXXVI 
HISTORICAL RATE-OF-CHANGE CURVES 


For the curves of historical data we find a type of rate- 
of-change chart which is fast increasing in popularity and 
general use. Though it involves the logarithmic projection of 
scale, yet that fact is so completely camouflaged by the rulings 
of the chart (in ‘“‘decks’’ as described in the last chapter) that 
we need no longer apologize for its use in a popular publication 
nor attempt to explain it in conferences. We merely murmur 
something about its being a truer picture of fluctuations and 
let it go at that. Ifthe other chap does not understand—and 
this includes chief executives and officials—he at least realizes 
that he ought to understand and enters no protest. In short, 
this chart form has already reached the stage of notoriety in 
which it need no longer skulk about in laboratory corners, but 
can parade in public with a slightly exclusive, but very ef- 
fective manner. 

The peculiarity of this chart is that it has a logarithmically 
projected scale along one axis only, the vertical or y-axis. 
Its x-axis 1s innocent as a new born babe of any such develop- 
ment—that is to say, its x-axis is projected arithmetically. 
And for this reason the chart is often, in charting office par- 
lance, somewhat crudely but tersely called ‘“‘semi-logarithmic.”’ 
Technically, of course, it can only be described as “‘x-arith- 
metic, y-logarithmic.” That the form is appropriate, however, 
for historical data, may be seen if we recall the law of organic 
growth in which it was stated that growth by uniform per- 
centages took place at uniform intervals or periods of time. 
Indeed, as the law of organic growth essentially deals with his- 
torical data, that is, data during various points or periods of. 
time, we may consider that chapter a discussion of the general 
theory of the historical rate-of-change chart in particular. 

The student may be surprised that time, in itself a natural 
phenomenon, should defy the law of organic growth, Eminent 

416 


HISTORICAL RATE-OF-CHANGE CURVES 417 


THE WORLD'S PRODUCTION 
Estimated gold, iron, coal and cotton production and population 
World. Specified years, 1800-1919 
(Note:- Gold figures are anmal averages for ourrent decades before 1900) 
(Source:e U. S. Statistioal Abstract) 


4 breed 2 : 2° 10 1 a e Es 
4 . 
cate, 7 8 Bo ne ees OE Sele See eee 
Lara Wil tin coh el a Oy a eee Meg eae 
t 
ee 
Coal A | CH cal a © Noon 
chert? 3 Pee mete ee oS eee 6S) 8") Be eee ae 
ey 7 Saa33 
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Fig tron $2 8 2 ® 5 C e Ses mI GEGe 
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: 
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a Pe ea eee ae wa Se CSO Pave sees 
Phe a? a Sele wae edane 


[] 2e 


ESET SUS oes ee TST 
aD, 


1800 
820 
$0 
40 
60 
60 
70 
80 
90 
1900 
06 
10 
13 
16 
17 
1919 


Fig. 348. Long-Time Series of Economic Data. 


engineers with exceptional mathematical ability, have ques- 
tioned the logic of charting time arithmetically with logar- 
ithmically projected dependent variables. The answer may be 
found in the very close approximation of the logarithmic series 
to the arithmetical one through small ranges. In the whole 
history of time the origin is so infinitely far removed and the 
range of known history so extremely minute (comparatively 
speaking) that there would be no appreciable difference in the 
resulting chart were the x-scale plotted by either method. 
While it is certain that for modern times the two would coin- 
cide, it is furthermore impossible to fix the true origin of time. 
For zoological and palaeontological charts the birth of the 
moon, or some similar event, might be a useful zero-point, but 
in business and in historical charts in general the result would 
still be the same. In other words, we may consider the his- 


418 CHARTS AND GRAPHS 


VIOLENT-DEATH RATES 
United States, 1900-1920 
{Sources:- For Homicides, suicides, total accidents and automobiles, the Census Reporte; 
For Lynohings, the Tuskeegee Institute; for street accidents, Dept of Health, N. Y. City.) 
(Note:s Figures in parenthesis are annual averages for 1901-1905.) 


aan 2 o an © 2 & 
Streetactiemts 2 F £3383 85888 38 S383 R eS 
atom Se 
New York City CO . Oi +0" .OP16s ON kA VO) tee 0 eG OPO ee One op ae 
re Be ar ae 
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9: 8 4 SU sem eens STS ey eee Gas 
Railroade SS SNS Si eS AS Sat OS ees 
~ 
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Total accidents 4 a Se Se eR Pai ea) con urate tole are an cee 
F a a lmsten Pat se OE ES ee gue ON De re Bei ee 
and unknown i 3 o © oo Oo Fe & o oO 
= 
rey a © 0 oFCnN 0 mM ofA A 
Suicides a 3 & 6 6 §$ 6 8 6S 6 ¥ HA 
- 3 en ee ee er ee, | 
S 
a ry + .0© ®9 © BA HW DMO 
° ° y: Cia SRR elo Real neers! ues OES 
Homicides a a o 0 8 © 0 F&F ~& © F & © 
~ 
BRA OD OD Rt OO) hac ep cl ee eae eye 
= a 28 Oo & & t& © & © © e- wo 
lynobinge OR SAMAR 8 6 SOR se Sie Se aie SUR Te a 
100 ———— 


HT THIN 


UA 


AL LT 


qj 
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mate (e€a 100,000 popuratian) 


ANTE ATT tT 


AA TM 
TA NTE UN 
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Ea 
12 Eel 
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eatkes 
——— 
“08 =s 
zeal ea [Cod es 
in POS BES 
18°88 BsSuS 68 Bs Sse SN Se aS AS eee 
m2 ERA oad a is g 
Fig. 349. Short-Time Series of Economic Data. 


torical rate-of-change chart to be really logarithmically pro- 
jected on both axes, with the x-axis scale range, however, so 
short that it appears to be arithmetical. 

In the construction of historical rate-of-change charts, the 
same principles apply (with the exception of the logarithmic 
scale on the y-axis) as in the construction of historical amount- 
of-change charts, to which several chapters have been devoted. 
A standardized form is useful when many charts are being 
made, most of the marketed forms described in the last chap- 
ter being useful for this purpose, as they contain no ordinates 
or vertical rulings and can easily be provided with the latter 


HISTORICAL RATE-OF-CHANGE CURVES 419 


to suit each case. The field should not occupy more than 
half of the page (on an 84 by 11 basis) in order that data can 
be entered in full. The same position of the field in the lower 
right hand corner should be maintained, when the charts fit 
together to form a series, in order that they may be closely 
overlapped either horizontally (to show one continuous curve) 
or vertically (to show seasonal and cyclic fluctuations). The 
same problems as to plotting points of data at unequal time 
intervals arise and they must be settled in the same way. In 
general the historical rate-of-change chart is but a duplicate 
of the historical amount-of-change chart, modified in regard 
to its y-axis scale and the plotting of its dependent variable 
data. , 

Not all the historical amount-of-change charts, however, 
can be duplicated in this way. With the simple curve, there 
is of course no difficulty (either in its usual or silhouette form). 
But in the Zee-chart the cumulative is not profitably copied; 
indeed it is rarely of any use to plot historical cumulatives (at 
least when these are for limited and repetitive cycles or periods 
of time) logarithmically. The bar-charts (except silhouette 
bar forms of curves) should never be plotted logarithmically, 
as by their nature they suggest and imply a significance in their 
heights, and the logarithmic chart, as you know, is without 
significance in the height of its curves.1 


1Jt is here perhaps best to warn the student’ against too confident use of other 
chart methods for the purpose of showing rate-of-change fluctuations. In previous 
chapters the series of chain-relatives or chain-percentages has been discussed and it 
has been pointed out that the chain relatives are the anti-logarithms of the logarithmic 
differences between successive items in a historical series. As the rate-of-change 
chart shows by the fluctuations of its curve the amount of these logarithmic differences, 
it is easy to see that the fluctuations of a rate-of-change curve do not represent the 
chain-relatives or chain-percentages, but represent the logarithms of these chain- 
percentages. Attempts have therefore sometimes been made to present graphically 
the chain-relative series itself. A little experimentation however will show that the 
method is not for most purposes useful. To be strictly accurate, the chain-relative 
series must not be shown by a single continuous curve but by a number of successive 
curves in which each plotted point is connected with the previous ordinate at its 
intersection with the 100% line. The result is a wholly disconnected picture and is to 
some extent liable to wholly meaningless changes of form when the time units of the 
data are shifted or changed. Nor is it possible to present a more connected story 
by joining each of these disconnected curves and plotting these new points the proper 
distance above or below the last preceding points instead of above or below the 100% 
line, for this would amount to a cumulation of the chain-relative series, additively, 
when the series properly cumulates only by multiplication, and it would result in a 
curve in which no uniform scale is possible for the dependent variable, and in which 
points at the same height upon the paper have different meanings or values at different 
portions of the series. Except in very special circumstances the rate-of-change curve 
chart is the proper method for showing the rate-of-change of fluctuation in data. 


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HISTORICAL RATE-OF-CHANGE CURVES 411 


Where the charts are to be used popularly it is well to go 
to considerable pains to lessen the misunderstanding of the 
chart by people who do not understand its logarithmic nature. 
Some misunderstanding is bound to arise, but one of the sim- 
plest and most effective measures is to position the curve (where 
one only is shown) at about the height upon the paper at which 
an amount-of-change curve with similar fluctuations, would 
appear. Thus if your curve covers a range on the y-axis of let 
us say from the 100% point to the 200% point, the highest 
point on the curve is obviously double the absolute value of 
the lowest. Now if we cut off our chart-field at the 50% 
point, then clearly we will have equal distances between the 
bottom of the chart, the low point on the curve, and the high 
points on the curve. Now when Mr. Average Man—and it 
may be Mr. Average Congressman and seem very important 
to you—picks up this chart, he goes through the following 
motions: “‘Ah, one of those damned curves—what the deuce 
does it mean? Oh, profits on rotten meat. Well, I see they 
doubled during the war. I can understand this thing easily.” 
And then he drops it again. He does not notice that the bot- 
tom of the chart registered 50% and not zero on its vertical 
scale, but that does him no harm. He has at least seen one 
thing truly, the relation between high and low points. 

In business reports it is always a good plan to present both 
rate-of-change and amount-of-change curves simultaneously 
for all important historical data. The two charts of the same 
information should be face to face so that the reader is con- 
fronted with both at once, and cannot mistake the rate-of- 
change form or judge quantities by its curve. Nor is this 
useful merely to avoid mistakes on the part of readers uncon- 
versant with rate-of-change paper. There is often a real use 
for quantitative analysis of the data, which can be seen only 
from the amount-of-change paper. The rate-of-change chart 
is indeed for most purposes more effective, but it fails wholly 
to give any picture of total sizes or quantities involved, and 
is not the panacea which some of its enthusiasts would have 
us believe. The whole truth about a series of data requires 
not only the rate-of-change but also the amount-of-change 
method. 

In more scientific reports and in records which will only 
be used by those who cannot misunderstand them it 1s often 
useful to combine the two types of curve for the same data 


422 CHARTS AND GRAPHS 


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®8—RATIO CHART 


SHOWING RATE OF CHANGE A—INCREMENT CHART 


SHOWING AMOUNT OF CHANGE 
Fig. 351. Comparison of Rate-of-Change and Amount-of-Change Curves. 


The chart shows the growth of one dollar (lower curve) and six dollars (upper 
curve) at compound interest.—Permission of Mr. John Wenzel. 


upon a single chart, superimposing one upon the other. The 
simplest method of doing this is to plot one curve for the 
natural numbers and another curve upon the same chart, with 
the same arithmetical rulings, for the logarithms of the data. 
The first or natural number curve can be labelled Y and the 
second or logarithmic curve can be labelled Log Y. If the 
paper used for the chart is fairly translucent, it is not neces- 
sary to look up the logarithms in plotting this second curve, 
but a sheet with heavy logarithmic ruling can be placed 
underneath the chart and the two sheets held up together to 
the light while the logarithmic curve is plotted, from the 
logarithmic ruling on the lower sheet, which are not intended 
to appear upon the chart itself. The slide-rule will serve 
equally well, if its scale is appropriate, an engineer’s scale 
being used for the arithmetic curve and the slide-rule for the 
plotting of the logarithmic curve. A clearer method of show- 


HISTORICAL RATE-OF-CHANGE CURVES 423 


ing the distinct nature of these two superimposed or combined 
curves than by merely labelling them Y and Log Y, is to trace 
or draw in a small portion or zone of typical logarithmic ruling 
immediately around the logarithmic curve, thus showing that 
the latter has been cut out from a logarithmically projected 
chart and inserted upon the arithmetical chart. Needless to 
say, the combination of these two wholly different types of plot- 
ting of a curve should be made upon a single chart only when 
that chart will show a single series of data, for obviously several 
curves brought together in this way would become confusing. 
No discussion of the historical rate-of-change curve is com- 
plete without mention of its value in forecasting, The use of 

TRADE UNION MEMBEXSHIP OF THE WOKLD 

Number of menbore in 20 countries 


1910-1919 
(Source:- International Labor Office) 


a 


[oy Toy to) : ) 

2 EE ee ie ee lea 
Number ey ee SD o a ro) y Sy 
of OT i i ol a Paeae sei e 
Members o a me Q Q ~ ce Cy 
50,000,007 ——, ——; —> ——— FF 
f ’ 

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7 ol RS Rea ER cad 
OS a a Fea ae Dal aed 
Eas See ae a Pe. ee 
a (ii ai cel 
20,000,000 fees ee ee ES ee es 
(aes) Be Ss oe Ee ee eee ees Bas 
15,000,000 a ee eas (Se ee 
eS | ae re ee Se 
ey ord BS Sa nn Ra) ae a 
i eS ee ee Ba EES a eae 
BT 1000 00° (ol a ee [eae ee eee [es Serie ee a 
Fe ees SaaS Saeed Ser TT 
“0 Di) eee as ed a Ba eae 
eee aes wf | fans ORS Ca 
6,000,000 z . = a Ses eee 
g a > es = a a a p= 4 3 ‘On m 
Ce ee ne eR RE ee Se 


Fig. 352. Interpolation and Extrapolation. 
Compare the extrapolated forecasts with those in Fig. 175. 


the logarithmic projection in general for correlation studies 
has already been noted in the last chapter. But there is a 
very special usefulness of the rate-of-change chart in predicting 
from the course of past events, the probable course of future 
events. The curve is merely extrapolated to the ordinate of 


2 


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pee Ul meee | es oe es ok, SMe Ore Oe, i a a ee a hy a ¢ aR i, 4 


424: - CHARTS AND GRAPHS 


the desired point in the future and the intersection point is 
read from this extrapolation. The degree of success or reli- 
ability achieved in such methods of prediction depend of 
course upon the faithfulness with which the future develop- 
ments follow the course of the past record. But an estimate 
of the probability of this can often be made from an inspection 
of the past course of the curve itself. The method is only sta- 
tistically indicated where the curve of the past shows very 
regular and uniform trend. Through a combination of causes, 


it often happens that these estimates can be more reliably 


and usefully made by extrapolation of the rate-of-change curve 


Eipeacamumes Ly 


Millions of Barrels 


Scale of Increase 


100 
ION 1912 1913 1914 1915 1916 1917 1918 1919 1920 1925 1930 
From Joseph E. Pogue's ‘‘Economics of Petroleum.” 
Fig. 353. Careful Extrapolation. 


A straight line (dotted) oe been fitted to the curve and projected ahead ten years 
to give forecasts. In 1930 the forecast is about 1250 million barrels as against 
800 millions indicated by extrapolation on amount-of-change paper (see Fig. 176). 


than by extrapolation of the amount-of-change curve, for not 
only is the trend of the past often more uniform upon the for- 
mer, but also in general it is true that the phenomena which 
show uniform rate-of-change in the past can be more relied 
upon to maintain their trend than the phenomena which show 
uniform amount-of-change. 

The historical rate-of-change curve is indeed one of the 
most important instruments in the treatment of ordinary 


it opens up to you ; 


« 


CrarTerR XXXVII 


LOGARITHMIC FREQUENCY CURVES 


Hamlet: Do you see yonder cloud that’s almost in shape of a camel? 
Polonius: By the mass, and ’tis like a camel, indeed. 

Hamlet: Methinks it is like a weasel. 

Polonius: It is backed like a weasel. 

Hamlet: Or like a whale? 

Polonius: Very like a whale. 


This same doubt and debate arises over the shape of every 
frequency curve. Such curves commonly present the widest 
variety of shape and contour. It 1s indeed possible to con- 
ceive of frequency data for almost any curve which may be 
imagined. Some attempts have been made to classify these 
various forms and divide them into a few typical groups. 
These classifications rest upon the relative positions (along 
the x-axis) and magnitude (or amplitude) of the peaks and 
valleys in the curve. They are restricted to simple curves, 
presenting not more than one peak and two valleys (strictly 
speaking, half-valleys) or one valley and two peaks (half- 
peaks). Where more peaks or valleys occur in the curve, and 
these cannot be smoothed out by applying a process similar 
to the moving-average or total process, it is rather assumed 
that the curve is not simple, but compound, being really a 
combination of two or more separate simple curves. The 
breaking down of a composite curve into simple curves is a 
problem of advanced and difficult mathematical steps, super- 
ficially not unlike harmonic analysis, and cannot be discussed 
here. Outside of the engineering and scientific fields, these 
curves will be rarely met and the student of business and eco- 
nomic statistics need only acquaint himself with the treat- 
ment of simple frequency curves. We shall first consider these 
simple frequency curves as they appear on amount-of-change 
plotting paper. 

426 


, 


LOGARITHMIC FREQUENCY CURVES 427 


While no classifications have proved very useful, the best 
which has so far been formulated, and one of the simplest, is the 
classification of Professor Yule.! Yule finds four common types, 


VITAL SUPERIORITY OF THE FEMALE 
Percentage Excess of Male over Female Death Rate 
England and Wales 
1851-60 and 1901-10 
(Source:- Reports of Registrar General of England and Wales) 


1901-10 
2861-60 


Years of Age 


Fig. 354. Compound Curves. 


which he calls, respectively, the symmetrical curve, the moder- 
ately asymmetrical curve, the extremely asymmetrical or J-shaped 
curve, and the U-shaped curve. These names are self-explana- 
tory. Inthe first or symmetrical curve, there is a single peak, 
from which on both sides the curve slopes away symmetrically. 
The salient facts about this curve are its width or spread 
(‘range’), the height of its peak ordinate, and the relative 
heights of its ordinates at regular or certain irregular (“quar- 
tiles,” “decils’’, or “percentiles”’) intervals away from the peak 
ordinate. The last detail would tell us whether the peak was 
narrow, indicating great concentration of the observations 
about that point, or wide, indicating a scattering or “disper- 


1Cf. Yule, Theory of Statistics. 


al CHARTS AND GRAPHS 


sion” about the point, and there are mathematical methcds of 
expressing these details which may be found by consulting the 
statistical authorities. 


OUTPUT OF COAL MINERS 
Average Number of Tons of Bituminous Coal Mined by Pick-miners 
per 8-hour day in 118 mines, 17 states (all fields) 
United States 


(Source:- Ethelbert Stewart) 


Number Number 
of of 
Tons Miners 
Under 3 577 
- 4 913 
4-5 1251 
5-6 1492 
/- 6-7 1395 
7-8 1275 
6-9 984 
9-10 672 « 
190-11 462 
23-12 290 
12-15 172 
12 and over 336 


Fig. 355. ‘A Frequency Series Which Appears Slightly Asymmetrical. 


The second, or moderately asymmetrical curve, presents the 
same general form as the first, but its peak is no longer in the 
middle of the curve, being nearer to one end than the other. 
Consequently the curve does not fall away on both sides with 
symmetry. Here we have a new element which is statistically 
known as “‘skewness”’, the curve being skewed over to one side, 
and there are mathematical methods of describing more or less 
explicitly the degree of this skewness. 


In the third or extremely asymmetrical type of curve, we 
generally have but half a peak and half a valley, that is, the 
curve begins at a peak on one side and slopes down toward a 
valley, but the other half of the peak, the opposite slope, as it 
were, may be lacking. Where the opposite slope is present, it 
is very near to a straight vertical line and the curve is merely 
extremely asymmetrical. When it is lacking, the curve is 
called J-shaped and not only the lower or valley end, but also 
the peak end, of the curve may be “asymptote,” to the axis 
of the chart. By an “asymptote” is meant a line which, while 


LOGARITHMIC FREQUENCY CURVES 429 


approaching infinitely near to an axis, gives no promise of 
actually meeting it, no matter how far it be extended outward 
along the axis. 


THE DURATION OF MARRIAGES 
Divorces Classified by Years of Previous Married Life 
United States 
1887-1908 
(Source: United States Statistical Abstract) 


Number of 
Divorces 
(Totel = 900,684) 


300,000 


Number of pivorces 
nv 
8 
3 
Gees 
EBASS. 
ied 
esgce 


wo ° wo °o wo ° re) So wo fo} 
‘= a 2 v o 


Over 


Years of Married Life 
Fig. 356. An Asymmetrical Distribution. 


The dotted line is the same curve plotted upon a logarithmic horizontal axis. 
Note that, if this plot could be extended through the first open group, it would 
probably become approximately symmetrical. 


The fourth type, the U-shaped curve, is one which describes 
a letter U, being made up of a valley and two half-peaks. And 
though it may be a very sharp or pointed, V-shaped valley, it 
is perhaps more frequently found to be more or less flat, just 
the reverse being generally true of the other three types. And 
it might be added that this U-shaped curve may be found in 
both symmetrical and asymmetrical forms, according to the 
presence or absence of skewness. Yule, however, makes no 
subdivision of the U-shaped curves, as they are comparatively 
rare. 

It is now our purpose to show that these four more or less 
distinct types are often interchangeable forms, which can be 
evolved by different statistical and graphical processes, from 
the same original series of data. For this purpose we naturally 
regard the symmetrical type of curve as the more desirable 
form, and shall consistently refer other forms back to it. The 
reason for this, as has already been indicated, is that the more 


430 CHARTS AND GRAPHS 


regular and symmetrical a curve be, the safer and more trust- 
worthy would seem its use for interpolation and generalization. 
The symmetrical form is not significant only to the mathema- 


8KY CLOUDINESS 
Distribution of 3,653 Observed Intensities of Cloudiness 
Breslau 
1876-1885 
(Source:- Yule, Theory of Statistics) 


- oa Lt nm 
Observations oS Se 


2 SN ee ee ew 
EEE EEE 
i mbes. 

ee ee 

Pete ote ay rect ef 
ttt te 

2 1500 Se Prt Te 
EM ily cia) as) TRL WS: (ae eS EY 
: SS a HO 
$ Sao 
ra bi woe a ee 
Pe re OE Sie) SR hs ER Ya 
aden (i) i Tas GS LET, ke es TL 
bo XG deed, oh ea gh oda ee 
Oy ap he hee eee ed 
oy Nh Bal 1 | A Es 
ee 

Ss OS DS RY Hea en 
aaSSSSaSee 
Galea he ia ie aa ae 

a is es = a 


eo Nn bel vv wo o 2 a iS 


(Least) Degree of Cloudiness (Most) 
Fig. 357. Yule’s Example of a U-shaped Distribution. 


tician who will seek a general equation for the phenomenon 
described on the chart, but it is also more significant to the 
layman, who, having once seen the symmetrical curve, will not 
so easily forget it. And at this point we enter the subject of 
the logarithmic projection of the scales. 

To begin with the second, or moderately asymmetrical form 
of frequency distribution, we have to note that its curve can 
often be made symmetrical by plotting upon a logarithmic 
scale. Plotted on rectilinear co-ordinates, that is, on the 
amount-of-change plotting paper, the right-hand “‘tail’’ of the 
curve, that is the slope toward the half-valley at the right-hand 
side, is generally longer than the one at the left. At the same 
time, the values of the independent variable, that is, the values 
of the points along the x-axis, are larger at the right-hand end 
of the scale than at the left. So it is obvious that a logarithmic 
projection will shorten this part of the scale in comparison with 
the rest of the scale (for the log projection always condenses the 
larger values). The result is to shorten the longer tail, often 


LOGARITHMIC FREQUENCY CURVES 431 


enough to produce absolute symmetry in the two slopes of the 
curve. Nor does this process need to be wholly empirical, for 
the student will soon learn to detect in advance the curves 


COLLEGE SALARIES 
Salaries in American Colleges erd Universities 
Including Public and Private Institutions 
United States 
1920 
(Source: G. S. Bureau of Education) 


° ” x c ad 
” Lcd rol 
Assictonte Mo B for} 
2 oO Cal Pe ooo fn 
nn 
Instructors 3 BY z 2 a 
wo wo 
bd ° cal wow OO NN et 
Oo nx a cond 
Acsistant Professors S 8 2 3 2 
: % [2-8 § ga" 
Associate Professors i 2 ee bea) 
a 5 Sse See pe noer amor ¢ Soinens lay 
Full Professors rr 8 S 7 8 FFA 
“ Nn . cal 
Presidents, Deans = 3 8 2 3 iS 2 tht Sa es Ea OO ee 


and Directors 


MATIC 
BBA SiSs 
wanna 


Number of Persons 


wo i") 2 © ww Ow DO HWW wwWoow rw 
§ g = € £ & BEEECE Scecctecs 
= = = a rs _ Nn om te B83 SFR ganas 5 


Dollars of Salery 


Fig. 358. Six Moderately Asymmetrical Distributions. 
Note that these have been made more symmetrical as to their sides and more 
rounded as to their peaks by means of logarithmic scales. 


2 See Fig. 36] on p. 435, 


5, ot Rae ee iin 


which, by this treatment, may become symmetrical.3 The suc- 
cess of the method depends, of course, upon the nature of the 
independent variable, whose values appear as calibrations or 
scale-figures along the x-axis of the chart. 

From the above it is clear that it would be impossible to 
apply this method directly to data in which the independent 
variable includes both negative and positive values and crosses © 
through zero. When zero is not a limit of the range, but 1s 
included inside the range, it would be necessary, if the method is 
to be applied, to change the variable to values which are wholly 
positive or negative. ‘This can easily be done, if the zero- 
value proves on inspection to be purely nominal, arbitrary, or 
relative, and to have no real zero meaning. Thus if the series 


432 CHARTS AND GRAPHS 


RENT INCREASES 
Humber of Families Reporting Increases of Rent 
Government Employees, Washington, D. C. 
Year Ending Oct. 1, 1920 
(Source:- Monthly Labor Review) 


0 1,983 
‘2-9 83 
10 - 24 648 
25 - 49 480 
50 - 7% 142 
75 - 99 42 


100 and over 29 


Fig. 359. The Independent Variable is Measured from an 
Arbitrary Zero Point. 


be so arranged as to show deviation from its mid-point (median), 
mode, mean, or from some other particular value, the devia- 
tions, being measured above or below this point, will show in 
the table as positive and negative stubs, and the point itself will 
show as zero. By merely adding to all stubs the true value of 
this point, we return the data to its primary form and wipe out 
the false zero-value. In other cases the zero-value, though 
arbitrary, cannot be so easily given its true value, and the work 
is more difficult. When zero is a limit of the range of the data, 
and is actually met in some of the data, the log-chart can be 
used, but of course the values of the curve in the first interval 
(next the zero point), cannot be plotted, as that interval becomes 


+See Chapters XXVII and XXVIII. 


infinitely long. In the same way, the final class or group 
cannot be plotted where it is indeterminate, for it too is 
infinitely long. 

In general, the same considerations apply to the possibility 
and usefulness of the logarithmic projection of the x-axis scale 
as are applied to the logarithmic projection of the y-axis scale, 
already discussed in the foregoing chapters. The logarithmic 
projection would seem appropriate whenever zero is an infinites- 
imal limit (approached but never reached) to the independent 
variable. Such a condition is inherent in the nature of the 
phenomenon and can be detected immediately therefrom. 
Most economic data are susceptible to the process of log 
projection. It is noteworthy that each of Yule’s examples of 
the moderately asymmetrical curve can be plotted on logarith- 
mic paper and made symmetrical thereby. Human beings, for 
example, cannot have a negative height, nor a zero height; 
communities cannot have zero populations; manufacturing 
establishments cannot have zero employees; nor farms, zero 
acreage. Frequency series of such phenomena, classified as to 
their sizes, seem to call for logarithmic projection a priori. On 
the other hand, profit or loss can be negative, as can net worth 
and balances of all sorts, including “stock” or “fund” data, 
time and space dimensions with reference to particular points, 
and for such data the log projection would ordinarily be both 
impossible and meaningless. 

Whether or not the vertical or y-axis be given a log projec- 
tion is relatively unimportant. Ordinarily when the x-axis is 
so plotted, the y-axis can be also, and the resulting curve shows 
a more rounded, less pointed peak. This may or may not be 
desirable. For identification with the normal curve described 
in the next chapter, perhaps more often the arithmetrical pro- 
jection of the y-axis, even with the log x, is desirable, but it may 
be that the identity will be established only when both y and x 
are logarithmic. 

The third, that is, the extremely asymmetrical or J-shaped 
curve, presents two possibilities. If it is extremely asymme- 
trical, but has two “tails” or half-valleys about its peak, it may 
belong in the same category as the second or moderately 
asymmetrical curve. The difference in the degree of skew, 
which seems so much more violent in the third than in the 


LOGARITHMIC FREQUENCY CURVES 433 


0) 


4See Fig. 356 on p, 429. 


434 CHARTS AND GRAPHS 


ee ge a ‘oe 
LENGTH OF WORDS 


Distribution of 10,000 Words by Number of Lettera in Them 
(Source:- Bowley, "Elements of Statistics”) 


} oy) | the) 
ec LGae et On Oe Cer OoE ts Clee Ole OES 
wo a fez] Le] oO wo R 
Number x 2 eS Be n @o o oO wo wv N nd 
of Words - Ps = 


Number of Words 


v8 if ie ig eae Bs 
Arithmetical 
RS Hae a LR a VS Wa Ved A ad TS Wa 
5 Gr Te Or Selo 25 G14 RUG 
Logarithmic 
Number cf Letters 
Fig. 360. A Moderately Asymmetrical Distribution which the 
Logarithmic Scale Has Not Made Entirely Symmetrical. 


second, may be found wholly due to the greatly extended range. 
Thus if we classify farms by acreage, our table may include 
farms of less than three acres and farms of more than a thousand 
acres. Here the range is very great, being through three 
logarithmic decks, andthe arithmetical projection has obviously 
brought the log mid-point very, very close to its left end (and 


“: be 
f 


LOGARITHMIC FREQUENCY CURVES 435 


with it the peak, resulting in extreme asymmetry. The height 
of human beings varies through no such great range and hence 
shows, when arithmetically projected, only mild asymmetry. ° 


SIZE OF PARYB 
United States 
1920 
(8ource:- Consus) 
Average Humber of ooo © ° . 
Yarn® jn eecn S325 3 € = z § 7 
Single-aore sub-group “209 2 Ss oe i 
in each Class 
gesz2 3 2 A 3 § 
fotel Sunder of Purne SGEL SL 3 = s 2 3 
fp eeoh Clase 3s 8 3 é 3 3 + s + = 
AAS 4 = <= <= =? 
aa = 
cc 2 
“ sate Ss £ 8 z 
aes Limite . : e 
(in Acree) $n34 3 8 e 3 8 g 


Wunder ef Fearne 


3 10 20 60 100 175 260 600 1,000 
Humber of Acres 


Fig. 361. An Extremely Asymmetrical Distribution Made Symmetrical 
by the Logarithmic Projection. 

Note that the left “tail” of the lower curve has been extended through the open 

class (zero to two acres) as a straight line, and therefore obscures the symmetry. 


Large ranges are extremely common in business and economic 
statistics, and the data will nearly always fly storm signals 
indicating the need for logarithmic projection. ‘Thus the usual 
classification of cities by sizes runs through intervals with 


6 SIZE OF STRIKES 
43 Nunber of workers involved in strikes 
United States 

1916-1921 
(Source:- Monthly Lakor Review) 


Classes: No. of workers 1-10 11-25 26-60 61-100 101-260 251-600 601-1000 over 


No. of equiv. sub-classes 1 1.5 2.6 5 15 25 50 
Class Total 197 345 412 413 395 348 238 264 

1916 Clase Average (197) (230) (164) (82.5) (26.3) (13.9) (4.8) 
Class Total 164 296 341 358 358 284 193 286 

2 1917 Class Average (164) (197) (As6))" CES) (28-9)! Ges (3.86) 
a Clase Total 143 268 334 344 371 278 141 216 

2 1918 class Average (143) (175) (133) (68.8) (24.7) (1142) (2.82) 
% Class Total 170 279 333 382 465 339 205 376 

5 1919 Class Average (170) (186) (133) (76.4) (31.0) (13.5) (4.2) 
5 Clase Total 160 299 313 326 341 262, 136 196 

1920 Class Average (160) (199) (125) (65.2) (22.7) (10.5) (2.72) 
Class Totel 219 288 252 214 216 153 101 140 

1821 class Average (219) (191) (101) (42.8) (14.4) (6.13) (2.02) 

Approx. number of workers "6" Bree "36" May (ot "25" "S76" "750" 


Too 
Oo oO 


Fig. 362, The Double-Logarithmic Projection is Best. 


LOGARITHMIC FREQUENCY CURVES 437 


vatious division points, such as 1000, 2500, 5000, 10,000, 
25,000, 50,000, 100,000, 250,000, 500, 000, 1,000,000. Who 
cannot see at once that these intervals are fierely ieand num- 
bers approximating, as closely as convenient, equal geometric, 
and not equal arithmetical, intervals? 

When, however, the curve is J-shaped, that is, has only one 
tail, another possibility creeps in. For it may be that what 
we are treating as a J-shaped frequency distribution is really an 
ogive, that is, a cumulated frequency series. The cumulation 
of a distribution which is symmetrical on an arithmetical 
projection is easily detected, of course, but the cumulation of 
other forms may be neatly disguised in the description of the 
series and pass for a time unnoticed. The behavior of these 
other cumulatives we will discuss shortly, but it should be borne 
in mind that the most frequent example of the J-shaped curve 
is an ogive or curve of a cumulated series. And when you 
meet a J-shaped curve, examine it first to make sure that it is 
not an ogive. If it is not, then the possibility remains that 
it is a very extreme asymmetrical curve, which is really not 
J-shaped at all, and has two perfectly good tails, but one of 
them is so very short as to be swallowed up in the peak.6 Thus 
Yule’s illustration of the J-shaped curve, being the uncumu- 
lated distribution of personal incomes in Great Britain, is as he 
himself says, not really J-shaped, but merely so asymmetrical 
that its lower portion has been swallowed up in the mode of 
the series. 

As to the fourth type, the U-shaped curve, rare as it is, it 
presents two or three obvious possibilities, one or another of 
which may serve in its analysis. In the first place, we must 
note that it may be merely an approximation to a distinct 

“yes or no” tabulation, its two terminal maxima representing 
the two alternatives and its intervening minimum the more 
infrequent compromises. ‘Thus a tabulation of eyes by degrees 
of blue or brown color might show many wholly blue and many 
wholly brown eyes with relatively few eyes of the various inter- 
mediate shades, if the whole matter of color be a resultant of 
the presence or absence of dominant color determinants. Yule’s 
illustration of this type of curve suggests a similar condition, 
being a record of sunshine and cloudiness at Breslau. If clouds 
be a local evidence of a falling barometer over a wide area, 


> See Fig. 362, page 436, where this has happened in the last year. 


438 CHARTS AND GRAPHS 


and clear skies of a rising barometer, since it is obvious that the 
air pressure can shift only one way or the other, we might be 
justified in expecting the local conditions to show few inter- 
mediate results. Closely allied to this is a second possibility, 
namely that in a particular U-shaped curve we really are not 
dealing with a simple curve, but with a compound one in which 
two opposite, extremely asymmetrical and apparently J-shaped 
curves have been combined. 

As athird possibility, we have to note that since a U-shaped 
curve is merely one of the first or second curves upside down, it 
is possible that by taking the remainders or complements 
(especially if the data be in p ercentage form) of the dependent 
variable, we can right it again, taking the data out of the fourth 
class entirely and throwing it into the first or second (or even 
third) class, there to be treated as above outlined. And just 
as the symmetry of the minimum-maximum-minimum curve 
may be effected by the logarithmic projection, so the maximum- 
minimum-maximum or the U-shaped curve may be similarly 
converted from asymmetry to symmetry. Mortality rates 
often show an extremely asymmetrical U-shape, which can be 
changed to a valley of perfect symmetry by the log-scale, giving 
a beautifully rounded curve with clearly emphasised variations 
for different racial and occupational groups of the population.§ 
And by subtracting the death rate in each age from the base 
of the rate, we obviously get what might be called a survival 
rate which, though less known, is a clear example of the second 
group of curves. 

It is not to be understood from the foregoing discussion of 
statistical and graphical methods of producing symmetry in a 
curve, that all frequency series can be made symmetrical by 
proper treatment. Sometimes the very failure of the series to 
be symmetrical is of prime importance and while we might by 
round-about methods produce symmetry, yet the inappro- 
priateness of these methods would be so great as to make 
symmetry meaningless. Nor is it true that all frequency curves 
will fall into one or another of the four classes mentioned. The 
point of what has been said in the foregoing discussion is that 
it is sometimes, indeed often, possible by very simple steps to 


°The Census Bureau in its Life Tables has published elaborate charts of these 
curves, but unfortunately through the use of arithmetically-projected scales, it has 


been obliged to break up each curve into three or four parts, each part with a suitable 
but different, scale. 


LOGARITHMIC FREQUENCY CURVES 439 


FEMALE MORTALITY RATES 
United States 
1910 
(In percentages ) 
Source:-- U.S.Census, 1914. 


a 
>) ro.) ist 2 ON Oar 
~~ o e a ix 2) © fs) 
mt og 2 * a 8 8 88 885 3 3° 8 ou © + + Onna 
cs wD nu cal Lal + cel Cal - ~) + roiea 
AAD 
Foreign 20° 8 
No MHS + re) rr) 
Pera . . . Be ee ae ee NSE PR 6 2 8 Tee eect ie 
i . : : . . 
© no 
Aas 
Nnee 2 a + + a ¢ & aoNnAsd 8 ° nm 0 ro) 
ative o on ix no fF MOM NANA a s o o Cc 8 oS Ba 0.6 
white o a a ‘: ness 7 davon’ 
a 


60 
50 


mal 
atin 
| i 
| 
aH 
| Uy 
g. CH 
Hy 


5 nies r= tt 
Sean feecuesaaeee ale 
SE SEs Vee ee rea oan 
21S ee | a 


Percentage of Deaths in Previove Year 


Yoars of Age 


Fig. 363. An Extremely Asymmetrical U-shaped Distribution Brought 
to a Beautiful Symmetry by the Log-scales. 


Note the emphasis given to significant irregularities, such as the increased 
mortality-rate at adolescence, among negroes, and its absence among the foreign- 
orn, 


attain the desired symmetry, and discover an underlying 
regularity in the behavior of the phenomena observed. To 
determine whether the symmetry is desirable and significant, 
and to interpret the meaning thereof when the symmetry has 
been secured, is indeed a task calling for experienced judgment. 
Furthermore, to detect the possibility of such symmetry 


440 CHARTS AND GRAPHS 


MORTALITY RATES 


The United States 
1901 and 1910 
(Source:= U. S. Census Bureau) 


0 rs) 8 88 Ae2k 2 gyag @ Qader v 
oo toa Ve) + RaAaOe 
z R a 
re ry A daw 
a 
© yy E © @ HD SMHE ORE GO 6 6 MH oOo a 
o a i a oom] Rn WwW eee or brann wo 
gs 3 3 o 3S FE eeayae ¢&t93 © 2AM B 
EY 
oH 3 ” aoa daa a 


se 8 @ 


w 
oO ° 
So °o 


60 


30 


Cs 
o 


~ iy) 
S cS) 
cA 

@e 


[ \ cea es 
—— TS asl EEA EEE 
8 
eae SESE SEE 


an 


Percentage of Deaths 
ix) a > 
ar 
= 
fame) 


fe 


ry 
@o 


SHEE 
See cctisaies 


PIN 
oh leeal OBialan 


2 3 4 6 Sire 6 14 5Q 40 50 60 
Age in Years 


° e'6 
Cy a n 


Fig. 364. Sameas Previous—Historical Comparison. 


Hf 
HH | 
Sette 
eae Menu 
os 
Pert 
Ser 
Mensson: 
ES 


80 100 


Deaths at Each Age, Shown a3 a Percentage of Those Living at the Beginning of the Prevtous Year 


ee 


LOGARITHMIC FREQUENCY CURVES 441 


requires imagination and familiarity with statistical data 
which the student will not quickly achieve. 

Some idea of the sign posts which indicate the appropriate- 
ness of the logarithmic projection has already been given. 
When the class or group limits which break the phenomena into 
a series are at widely varying and rapidly increasing intervals, 
soch as (0),11, 2, 5, 10, 20, 50,'100, 200, ... . (infinity) or 1 
month, 2 months, 3 months, 6 months, 1 year, 2 years, 5, 10, 
15, 20, 25, 30, etc. years, and similar arrays, then the geometric 


CONVENIENT GEOMETRIC INTERVALS 


Number per “‘deck” 2 3 (4) 5 6 7 8 10 
(powers per ten) 
Approximate Ratio Sree eel a (lee alee le Om eet ot mlaaer a tees 1.25 
_ Between Intervals 
Intervals 1 1 : 1: Wh, tle il, E 
3 2 Eales he 1.4 1.4 es 1525 
10 5 3 eS ae Be 1.8 1.6 
10 Se ome ae 3 PETES aals | OR: 
10. Ge 4.5 HD BP 2e5 
10. See eee Bee Ze || Bey 
10. 1 PS), 55. |) ze! 
10. bail i Sc 
10. 6.25 
Se 
10. 


Fig. 365. A Table of the Convenient, Nearly-geometric Intervals by 
which the Range Between Successive Powers of Ten May be Divided. 


nature of the progression is evident. But the law of organic 
growth applies in far more cases than those which wear this 
obvious marking. It may not be amiss to note some of Yule’s 
illustrations, since we have followed his classification of curves. 
As he says, the symmetrical curve is rare in economic statistics 
and his one example of it, relating to the stature (height) of 
groups of adult men, covers so small a range in inches (from 
58 to 76) that the logarithmic projection of the scale would not 
appreciably alter it. Hence we may conclude that while this 
particular series is practically symmetrical—on arithmetical 
projection—yet it may really call for a logarithmic projection 
because of the organic nature of human growth, and the arith- 
metical symmetry may be entirely accidental. This idea grows 
stronger as we observe that his illustrations of moderately 


AMERICAN ACCIDENT TABLE 
442 Duration of Temporary Total Disability 
A (95,388 cases per 100,000 accidents) 
United States 
1919 © 
(Source:+ Olive E. Outwater) 


MMOD EMNODONO SE DOF ONHHOPTHH OIG 
€ A RbSeeheereieeaheas* “=F 28 
Cases t 5 Sue aad } 
o 
Pe ; 3 bit 2:8 6.4.2 536 ££ 2 22.8. 2.82 2. Ee 2. 68.8 7h 
Bo 
° at ot 0 HT MOLCOATANDMAMHAMAANae 
i Period . ’ CSO ATE CONC ch CUBES LC SCR Car iC St Cust Oi a8 75 
© = Hehe OCS nie Seer sas eras 
MONTDOHOrFOMONWNDS 
R38 388 885 25888 
Cases ootonrwsunannan 
PS 
rr) >> 
Qa SES Re Co Cie eal 
Period SAE CR HD: S19} B=: CO OO set OE /8p at 
40,000 
35,000 ; 
' 
‘ 
' 
' 
30,000 +e 
‘ 
, 
800}—+ 
& 26,000 oO aay 
8 400;—+ 
% ‘ 
°o t ‘ 
200; — 
B 20,000 ‘ 
3 t 
agg t 
60 : + a 
16,000 40 \ 
hal 
\ 
— HHH AS HAH 
10,000 ‘ line 
9 © Coo Me Co 
re CEO ¢ Se Ml 
; * Ce ee bate l 
6,000 NK 
~ | So 
2 SN 
| ie: il LH L l AANA 
0 a = ss Piste. 
1 2 3 45 6 7 10 14 <—(Daye) 
(Weeks)——> 1 2 374 8 6 7620. 15 20! 26 
0 i) 10 16 20 2s 
(Weeks) 


Length of Disability 
Fig. 366. The J-shaped Distribution. 
(Shown by the dotted line and plotted upon the outer or arithmetic scales) 


becomes more rounded as logarithmic projection (shown by the full line and 
plotted by the two inner or logarithmic scales) is used, 


LOGARITHMIC FREQUENCY CURVES 443 


asymmetrical curves include distributions of the stature of ie 
and young men, in which the ranges are considerably larger 
(great enough to show appreciable difference between the two 
projections) and in which the curve becomes symmetrical on a 
log-« scale. The weights of the adult men likewise showed 
moderate asymmetry en disappeared on the log-x scale, 
again because of a greater range (from 100 to 250 pounds). In 
all these cases, the class or group intervals are even and regular, 
and yet from the organic nature of the phenomenon—human 
growth in height or weight—we could suspect the desirability 
of the logarithmic projection which is so successful in fact. 
The true significance of the success of the log projection in 
producing symmetry, when it does so, is that the proper units of 
growth are magnitudes, such as those by which we measure 
star-brilliancy or musical pitch, rather than increments; nature 
uses geometrical, not arithmetical, units. 

As has already been said and as may be deduced from the 
above, it is not always necessary that both axes should be upon 
the logarithmic projection. Frequently one has use for a chart, 


_the scale of which is arithmetically projected along one axis 


and logarithmically along the other. Thus we may note at 
once four possible chart-fields: the plain rectilinear co-ordinates 
or x-arithmetic y-arithmetic; the logarithmic or x-log, y-log; 
and the two semi-logarithmic, x-arithmetic -y-logarithmic, and 
x-logarithmic y-arithmetic. The discussion has so far turned 
upon the projection of the x-axis scale. With regard to the 
y-axis scale, the different projections obviously do not affect 
the symmetry of simple curves and we must base our selection 
of the proper projection either upon the nature of the 
variable to be shown and the apparent appropriateness of either 
method, or upon the emphasis or detail which we wish to give 
to certain parts of the curve, or upon mere convenience. The 
log projection of the y-scale always gives more pointed valleys 
and more rounded peaks than the arithmetical projection. 
Log-logs, or the logarithms of logarithms, have been mentioned 
in a previous chapter, and by their use still further changes can 
be effected in the contour of the curve. In short, the student 
will find ample means in these projections to study his data in 
various forms in the course of his analysis. 


CuapTer XXXVIII 
LOGARITHMIC OGIVES 


It is in regard to the ogive that the projection of the y-scale 
becomes important. It will be recalled from the chapter on 
ogives, that these charts show the cumulation of frequency 
series. It will also be recalled that the frequency series can be 
cumulated from either end, forming either a “‘more-than”’ or a 
“less-than’”’ cumulation. Hence, even on an arithmetically 
projected field we can always have two ogives for the same 
original uncumulated frequency series. If this series be a sym- 
metrical one, the two ogives will mirror each other on both axes; 
but if the original distribution is asymmetrical, they will not 
mirror both ways even when arithmetically projected. But by 
the use of logarithmic-scale projections, either on one or the 
other or both axes, we can sometimes produce mirroring again, 
a condition which often indicates that on such scales the curve 
would become symmetrical. For the treatment of the ogive, 
then, as for the simple curve, the proper projection of the x-axis 
is important. But in ogives we have, of course, only one 
maximum and one minimum, with the entire range of the series 
distributed between. Hence it is sometimes possible to secure 
in ogives what can never be secured in uncumulated frequency 
series—a straight line. The single exception to this is the 
J-shaped frequency curve with only one tail, and this curve 
will often, as has been said, be found to be an ogive in disguise, 
the cumulated nature of its data being not immediately 
apparent. 

Now just as symmetry is more desirable in general than 
asymmetry, so a straight line is in general more desirable than a 
curve. For it still further simplifies the significance of the 
chart, and it still further displays regularity of behavior in the 
phenomena. Hence in work with ogives, which are merely 
irregular curves, typically of an S-shape, the search for a 
straight line is legitimate. And at this point, the value of the 


444 


log-projection of the y-axis scale comes in. 


LOGARITHMIC OGIVES 


445 


For it may be that 


an ogive of very great curvature will straighten out into at 


Yearc 


1921 


1920 


1919 


1918 


1917 


1915 


. 
Total 


1,147 


1,262 


1,796 


1,650 


1,401 


2,063 


Number of Strikes 


DURATION OF STRIKES 


Number of Strikes ended in Specified Periods of Time or Less 


United States 
1916-1921 


(Note:- Strikes omitted because lengths not reported:- 1916, 332; 1917, 616; 1918, 464; 


1919, 301; 1920, 437, and 1921, 262.) 


(Sourece:- Monthly Labor Review} 
e a a one oO Nn ANMDROOY HH DO MH O 
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é o ° 2 2 bes oof ADM ate aw am 3 
Lal Ded wo oO - OD num mM nO OFb © © on an 
HA Ave eA Addt AA AAA a 
2000 i ale | [ | | 
1800 
| ie —— 
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1600 af 
i} ‘ 
2 2S il 
| 
1400 J 7 
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1200 
1 SY + 
a2 
=a 
1000 
+ a meet 
1 
wy, vy, 
800;— 
3 
800 
= + — 
400 
200 [ 
4 
= ii 
100 
| eas 
a AN AR Mm & A a 
or n OOPS RST Ch beet Pearls One aS SUNS aD 
Days: - A 
r) o t+ HOF ADAM 
Weeks:- = FRCS 
Months; - x cH) 
~ / Duration of Strikes 


Fig. 367. Ogives Plotted Upon Logarithmic Horizontal Scale. 


Compare Figs. 283 and 379. 


446 CHARTS AND GRAPHS 


least a close approximation of a straight line when the y-scale 
is made logarithmic. This possibility applies in general to all 
truly J-shaped curves. And when you have been unable to 
make a curve symmetrical, you still may succeed in making one 
of its ogives into a straight line, and so pull success out of 
defeat, and bring order where chaos was. 

A spectacular example of this close approach to a straight 
line on the part of an ogive is the “more-than”’ cumulative of 
the distribution of personal incomes in a community. Govern- 
ment reports show tabulations of the number of persons who, 
according to their income tax statements, enjoyed incomes 
between specified limits. By cumulating this distribution so as 
to get the number of persons enjoying more than each specified 
amount of income, there is obtained the data for an ogive 
which will pass through such excessive ranges in both variables 
as a dozen persons enjoying more than ten millions of dollars 
annual income and two millions of persons enjoying more than 
a thousand dollars of income. The curve of this cumulated 
series will, on arithmetically-projected scales seem asymptote 
to both axes, hugging them so closely throughout its length 
that were the chart-field as large as the side of a house, the 
curve would never leave the axes by more than a few inches. 
But plot the same data on logarithmic paper and the ogive 
comes so close to a straight line that one is tempted to ascribe 
its variations to errors in the data. This particular example 
not only illustrates the tremendous compression of large num- 
bers on a log-scale, but it also nicely exhibits the analytical 
power of the logarithmic method, for by its use the Italian 
economist, Vilfredo Pareto, was led to formulate a “law” of 
the distribution of wealth and income which was simply the 
mathematical expression of the slope of the straight line. And 
while Pareto’s law has not withstood the waves of debate which 
it has occasioned, the straight line on which it was based 
remains the best means of analysis of comparative income 
statistics for different communities. 

Another advantage in the logarithmic y-scale, and one 
which applies to all frequency curves, as well as to ogives, is 
that by its means, very dissimilar (as to amplitude or height on 
the y-scale) frequency curves can be compared. Thus when 
two series would on ordinary co-ordinates lie so far apart, the 
one high above the other, that they could not profitably be 
shown on the same chart without using different vertical scales 


PERSONAL INCOMES va 
Wumber of Persons Reporting Incomes in Excess of Specified Amounts 
United States 
1919 
(Source:- Collector of Internal Revenue) 


o fe o o 0 on © 20m HaD + ro 

@ ~* a3 Oo WwW a9 = nu < wm ove ao wo 

@o xa «a @ a Onw 7a wo trou On 
Number of Persons = Abeer ve am path caramal als ars, wx 

BWiste ea ece 86S ipo. iS 

. SPs 

ie] a 

Pca | Ei eee Pe Ae 


aun v2) 
ee 
Nie 
[| Baa A) ei) 
ne ae i 
Cian hee 
SSR 
ri te Ui a il la a as 
Se ieaiaiel Withernsea 
(40 e mag aaauee LN 
. all Sane = ae 
eo EE 
cy Sane i sat 
_- au 


' 
80 H 
60 CO 
8 eee ee ee eae Be 
eee eee aes 8 8 (es 


Dollars of Income 


Fig. 368. An Ogive Plotted Upon Both Logarithmic Scales. 


rs ee Aa ee eS ee ere re ee eee ee eee ee ee ee ee ee fe 
eet para oe Bicuest j 5 r at Bis’ ati 


448 CHARTS AND GRAPHS 


for them, it is often a very great saving in labor, as well as an 
assistance in analysis, to use the logarithmic vertical scales. 
The slopes of the various parts of the curves may be compared 
upon this projection and significance attached to parallelism, 
just as in historical rate of change curves. Commonly, the 
method is most useful when the two curves lie upon the same 
portion of the horizontal scale. Curves can also be made com- 
parable by the use of percentages in the place of the numbers, 
each value being turned into a percentage of the total of the 
series. This requires more computing, but is for some pur- 
poses superior to the use of logarithms or logarithmic -scale 
projection. 

In the construction of the logarithmic frequency curves, the 
principles laid down under amount-of-change frequency curves 
apply as to the positioning of the field. When the ogive is used, 
there should be room above the field of the chart for the original 
data to enable reading from the independent variable, and it 
may often be well to leave room to the right of the chart for 
derived secondary data in the form of readings from the depen- 
dent variable. As for other frequency curves, the selection of 
the independent variable is sometimes difficult and often merely 
a matter of whim, choice, or convenience. In such cases, 
particularly, both the original and the secondary derived data 
are useful, for you cannot be sure in advance which data you 
will ultimately adopt. There is, however, in the logarithmic 
frequency curve, little use for the staircase form of plotting, as 
the areas between ordinates have no significance, and we can 
limit the discussion of the logarithmic curve to frequency 
polygons and smoothed forms of curves. As to the logarithmic 
projection of scales, of course, the principles laid down in the 
previous chapters apply. And other things being equal, the 
logarithmic projection of the x- and y-axes of each chart 
should be upon a common scale, that is, with decks of equal 
size, whatever the calibration may be. 

The possibilities of application of the logarithmic frequency 
curve and the logarithmic ogive are very great and the student 
will soon discover that they exceed in usefulness for research 
purposes the ordinary amount-of-change frequency curves as 
much as the historical rate-of-change curves exceed the his- 
torical amount-of-change curves. 

We have mentioned population distributions as obviously 
calling for logarithmic projection. This is important in sales 


y \ m ' 


analysis and merchandising research. Rent statistics, pos- 
sibly an even better index of local buying power than income 
statistics, have been found, where they have been compiled, 
to behave as do the incomes. In building statistics, it has 
been found possible to set up normals of new building for each 
community on the basis of size of population, and the com- 
parison of the actual building with this normal affords a useful 
index of local business conditions, the entire analysis being 
carried out on logarithmic paper. The possible useful applica- 
tions of this form of chart are innumerable. In the engineering 
world it is in common use, being much better known than the 
semi-logarithmic, historical rate-of-change curves. Though 
less popular than the latter and perhaps less often required, the 
logarithmic frequency curve should play an important role in 
the business or economic research laboratory. 


LOGARITHMIC OGIVES 449 


. 


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ie AED el yh aha 


. aye 


pear HIS ae 


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ee Pee sa et ee aaleen = 
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i ’ & a greene , 


Ge 


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mesures i aaa den aah i 
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We: : 


a Na li a A he ee $ iy | ee ee oo et rr 2 
ia 2 : Pe ~ , 


CHAPTER XXXIX 
THE NORMAL CURVE OF ERROR 


The statistical authorities usually make a great to-do over 
frequency curves, for much statistical work of a precise nature 
involves the close study and measurement of frequency distri- 
butions. The fundamental conception in such work is one 
ideal or theoretical form of distribution which is known as the 
“normal curve.”’ The analysis is more often than not directed 
at the question of whether given distributions conform to this 
normal, and if not, how closely they approximate it. In 
graphics the question is whether a given frequency curve can 
be found to be identical or nearly identical with a corre- 
sponding normal curve. For reasons which will presently ap- 
pear, the discovery of this identity is always attended by re- 
joicing and relief, similar to the discovery that a historical 
curve conforms to the law of organic growth. For in each case 
a condition of meaningless irregularity has been replaced by 
the establishment of a definite and significant law. 

If you select at random a hundred men, provide them with 
yard-sticks, and let them measure the length of the city block 
in front of your house, you will be disappointed if you expect 
a unanimous report fromthem. Their measurements will vary 
through a considerable range, and “bunch up” most thickly 
about midway between the extremes. Pick your men for ex- 
perience and ability and you may expect the variation to 
extend through a shorter range, that is, the highest and lowest 
estimates will be nearer together, but the variation will still 
be present, with its bunching up at the mid-point. Provide 
the men with accurate engineers’ chains instead of wooden 
yard-sticks and you will still further reduce the range of varia- 
tion, perhaps down to inches instead of feet or yards, but it 
will still be present. The results are not changed if, instead 
of a hundred men, you use but one man, letting him measure 
the distance a hundred times over (provided he does not 


450 


THE NORMAL CURVE OF ERROR 451 


voluntarily repeat his estimates). As a matter of fact, all 
these measurements are approximations; the actual cniee 
itself has not changed or varied in the least. We are simply 
confronted with the human equation and its inevitable errors. 
And the interesting thing is that these mistakes or errors have 
been found to group themselves in a certain characteristic for- 
mation or distribution. In any sufficiently large body of ob- 
servations the scatteration or dispersion of items due to mis- 
takes or errors of observation, falls ever into or approximates 
the same characteristic form. Plotted upon a chart, this form 
is the normal curve, and the normal curve is therefore often 
called the curve of error or the curve of errors. 

Under the name of “the probable error,” artillery officers 
study the scattering of gun-fire, for no two successive shots 
from the best cannon in the world will hit at precisely the 
same spot. If you step outdoors and pick (without choosing) 
a hundred leaves from a tree you will find that while they may 
‘be of approximately uniform size, yet there will be minute 
variations of length or breadth in these leaves. Or glance at 
the stock exchange quotations and arrange the first hundred 
into a frequency series. In every case the form of normal 
curve will again be approached by the curves of your observa- 
tions. Hence the normal curve is often called the ‘“‘probable 
curve” or the curve of normal probabilities. 

In algebra every school boy knows the expansion of the 
binomial (a+b). The square of the'binomial is (a?+2ab+0?), 
Its cube is (a8+3a2b+3ab?+b3), its fourth power is (a4+ 
4a3b +602}? +4ab3 +)4); its fifth is (a5+5a4tb +10a3b? + 10a2bs + 
5abt+-b*); and so on. Note that these coefficients increase 
always as they approach the center from either end. If we 
were to plot them as a frequency curve we should again find 
a suggestion of the normal curve. Carry the expansion out 
into higher powers and the approximation becomes closer. 
If the expansion could be made indefinitely great, the curve 
of the coefficients would precisely conform to the normal 
curve and for this reason the statistical study of the normal 
curve is closely tied up with the binomial theorem. If in no 
other way, the normal curve could be mathematically com- 
puted by its means. 

Enough has been said to show the importance of the normal 
frequency distribution. Let us therefore attempt to describe 
its appearance. This is not an easy task, in as much as any 


452 CHARTS AND GRAPHS 


THE NORMAL CURVE 
Ordinates of the Uncumulated Series 
(Formula: y =e—3%’) 


(Nore: Values (ordinates) are given for parts of the range (abscissae) on both sides 
of the Median (origin), the units of measurement for abscissae being the Standard 
Deviation, and for ordinates, the Median.) 


Negative Side of Median (origin) Positive Side of Median (origin) 
Abscissa Ordinate Abscissa 
(x) (y) (x) 
0 1.000,0 0 
—0:2 -980,2 0.2 
—0.4 92351 0.4 
—0.6 .835,3 0.6 
—0.8 .726,2 2 0:8 
—1.0 .606,5 1.0 
—1.2 486,8 12 
—1.4 Ses: 1.4 
INE .278,0 1.6 
Sas 197,90 1.8 
—2.0 135,34 2.0 
—2.2 088,92 222 
—2.4 056,14 2.4 
—2.6 034,05 2.6 
—2.8 019,84 2.8 
—3.0 011,109 3.0 
—3.2 005,976 Shy? 
—3.4 .003,089 3.4 
—3.6 .001,533,8 3.6 
—3.8 -000,731,8 3.8 
—4.0 .000,335,5 4.0 
—4.2 -000,147,75 4.2 
aay 000,062,52 4.4 
—4.6 -000,025,42 4.6 
—48 000,009,930 4.8 
on) .000,003,727 5.0 
Fig. 369. 


change of scales upon the chart, along either axis, results in a 
change of its shape. Enlarge the horizontal scale and the 
figure of the curve is spread out wider, reduce that scale and 
it becomes narrower. Increase the vertical scale and every 
part of the curve is raised, making its peak much taller; dimin- 
ish the scale and it is flattened out. Hence the normal curve 
may have an infinite number of conformations. Always, how- 
ever, it has a hump at its horizontal mid-point where the fre- 


THE NORMAL CURVE OF ERROR 453 


quencies are thickest.* Hence it may be described as bell- 
shaped, showing a peak at the mid-points and a die-away curve 
or tail at each side of the peak, one at least of the tails being 
asymptote to (i.e. approaching but never reaching) the hori- 
zontal axis. . 

Now there are any number of possible curves which fit the 
above description but are not normal curves, hence the de- 
scription is not a definition. Curves may be rounder or flatter 
or more pointed at the top or may slope away at different 
angles from the corresponding normal curve or any similar 
normal curves. Hence it is important to be able to distinguish 
between normal curves and curves which are not normal. It 
is not enough to find in analyzing a given distribution, that its 
curve has a central peak and is symmetrical. ; The student 
should be familiar with the various forms of the normal curve, 
but it is not always possible, even for an expert, to be quite 
sure whether a given curve is normal or not, if he can rely only 
upon inspection. We need to compare the given curve with 
a precise drawing of the corresponding normal curve. And re- 
membering the infinite number of normal curves, it becomes 
dificult to select the proper one for the given distribution. 
Obviously the compared normal curve, that is the ideal dis- 
tribution, for a given series, is one which will have the same 
total area under the curve, that is, the same number of obser- 
vations or items in the series. But there is an infinite number 
of normal curves with the same inscribed area, differing from 
each other according as their peaks are tall and narrow or 
short and broad. So the proper normal for any series must 
be one which will intersect the given curve at certain points. 
Usually the standard deviation is used as the abscissae of 
these intersecting points. So we are led into a tedious math- 
ematical process, only a part of which is the computing of 
what statisticians call the standard deviation and all of which 
is more statistical than graphic. The comparison of a given 
curve and its normal is not easy by this method.! 

In the next chapter will be described an easy trick by 
which you can make such a comparison graphically, and learn 
whether the given distribution or series 1s normal or not, and 
if not, how closely it approximates the normal distribution. 


1The two ways of fitting the proper normal to a given curve, other than the 
graphic one in the following chapter, can be found in Yule, Theory of Statistics, pp. 
307-9, 


4 2 . be wk Lo Ase . PP Pe ~~ 7 -" Ae ee, A eh esr i ed 


re 
i 


CHAPTER: XL 
PROBABILITY CURVES 


Those readers who have understood the form of graphic 


‘legerdemain by which the adherence of a historical series to 


the law of organic growth is flashed upon the rate-of-change 
chart paper, will be prepared for a similar trick by which the 
adherence of a frequency series to the normal distribution is 
graphically shown. They will anticipate that the graphic 
form eliminates practically all computing and calculating 
from the treatment of the data, this computing having been 
absorbed once for all into the projection of the scale of the 
chart. The trick is very simple. It consists of plotting the 
ogive of the distribution upon paper which has been specially 
ruled off in such a way that the ogive of any normal curve will 
become a straight line upon it. 

The normal ogive, as you know, is S-shaped. If you 
know the ordinates of the normal ogive you know precisely 
how much to distort the scale for the dependent variable so 
as to produce a straight-line projection of the normal ogive. 

You need only select points equidistant along the vertical 
scale, read the abscissae of corresponding points on the ogive, 
lay off these horizontal values vertically (on the vertical 
scale), and shift the scale figures from the old to the new ver- 
tical scale. By doing this you have made the ordinates and 
the abscissae of each point on the normal curve alike and so 
of course the normal curve becomes a straight line. But any 
other S-shaped curve, any other ogive, or any curve at all, 
which is not a normal one, will fail to straighten out perfectly. 
And so at a glance you can see, by this chart, not only whether 
a given distribution is normal, but if it is not normal, how 
closely it approximates or deviates from a normal distribution. 

To plot a given ogive upon the probabilities projection of 
the dependent-variable scale just described, it is best.to turn 
all frequencies, 1,e, dependent variables in the data, into per- 


454 


THE NORMAL OGIVE 455 
Ordinates of the Cumulated Series 
(Note: Values (ordinates) are given for parts of the range (abscissae) measured in 
both directions from the Median as origin, the units of measurement for abscissae 
being the Standard Deviation, and for ordinates, the total of the series. The table 
can also be used to show fractions of the area under the normal curve (uncumulated) 
lying on each side of verticals (ordinates) from specified abscissae.) 


Positive Side of 
Median (origin) 


Negative Side of Cumulation 


Median (origin) 
“More than” 


“Less than” 


“Less than” 
“More than” 


Abscissa Ordinate Abscissa 


.500,0 
480,1 
.460,2 
440,4 
420,7 
401,3 
382,1 
363,2 
344,6 
326,4 
308,5 
.291,2 
.274,3 
.257,8 
.242,0 
.226,6 
211,9 
197,7 
184,1 
A711 
158,66 
135,67 
115,07 
096,80 
080,76 
066,81 
054,80 
044,57 
035,93 
028,72 
022,75 
013,90 
008,20 
004,66 
002,56 
001,35 
000,69 
000,34 


.000,159 


.000,072 
.000,032 
.000,003 


Me 


Fig. 370, 


500,0 
.519,9 
539,8 
559,6 
579,3 
598,7 
617,9 
636,8 
65544 
673,6 
691,5 
.708,8 
.725,7 
.742,2 
.758,0 
73,4 
788,1 
802,3 
.815,9 
828,9 
841,34 
864,33 
884,93 
903,20 
919,24 
933,19 
945,20 
955,43 
964,07 
971,28 
977,25 
986,10 
991,80 
995,34 
997,44 
998,65 
999,31 
999,66 
999,841 
999,928 
999,968 
999,997 


BS FES C99) SOS ALAS ES SSS ea tes ats 3 
MASON RNODARN SO DNDANA WHO © 


va ire ee. <i a fae Tee ie 


456 CHARTS AND GRAPHS m* 


centage figures. For the peculiar spacings of the probabilities 
projection are arbitrary and cannot be freely changed. The 
total of a series is the limit of its cumulation, and the cumu- 


TRE PROBABILITIES PROJECTION 
Oiagram showing the method of constructing 6 
Probabilities projection along the Y-axis. 


(Less-than eae 
latin oon bd 
sores BSS SANA Bane nomen mrmacerenons 
contege ee ee Scacoae 
RESSaAATORAAGD 


Essien ea cha kas oes 


of 
Frequencies 


Percentage 


Li 


(Median) 


2 
3 


Range (1, © Standard Deviation) 


Fig. 371. 


Showing how the dependent y-scale of percentage-frequencies is readjusted from 
the arithmetical projection which gives the curved ogive to the probabilities 
projection which gives the straight-line ogive. The frequencies must invariably 
be converted into percentages for this. chart. | 


lated series has definite limits (for its dependent variable), 
these limits being “no items” and “all items,” that is, 0 and 
100% of the total of the series. Hence the only common 
measure for all frequency series is a percentage one and the 
normal curve and ogive are both given in all tables in per- 


centages. The distortion of the dependent variable scale is 
made for values of these percentages, and you must not seek to 
shift the scale figures of the probabilities projection scale as 
_ you could an arithmetical or logarithmical projection scale. 
Your only way to alter the probabilities scale is to turn the 
percentages into values of your particular series, in which case 
the scale is restricted to this series, a fruitless if not danger- 
ous step. It is sufficient to turn the cumulated series into 
percentages before plotting upon the probabilities projection, 


| | 
| 


2 OD OHS a 
eee cease ‘e 
NAA deed , 
eee ones 


PROBABILITY CURVES 457 


Frequency 


on o 0 > DOD FM CANMINHO DOW! 
OFF el eee ir Oe Kgl 261 Ae Tene eLane serene 


Lae hea eS Je See Beet | o a a ° 
. . ° * ° ° * . 
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Nao 
7 ee * 
Are 4 
Ortret) 


. 


Fig. 372. A Less Useful Form in Which the Independent or x-Scale 
of the Range is Readjusted to Straighten Out the Ogive. 

It is less useful because the range-intervals must be turned into units of the 

standard deviation before plotting. Its sole advantage is that the frequencies 


need not be turned into percentages. 


The scale for the independent variable, that is, x-axis 
scale, may be projected either arithmetically or logarithmically. 
If a given series will straighten out upon the former, it indi- 
cates that its curve would be symmetrical upon an arithmetic 
projection; if the ogive straightens out upon the logarithmic 
paper the distribution is one of the asymmetrical types which 
are made symmetrical by a logarithmic projection. Hence the 
probabilities paper not only gives the approximation to the 
normal, but it also determines whether that normal be sym- 
metrical upon a geometrical or arithmetical basis. Chart 


CHARTS AND GRAPHS 


x 
gr » & Ke 6 9 NO eS 1. & a OG TTT 
° 
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8 
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i {| l Het Teo { 
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3 B oa a Owl i BE, i 
8 N 
: : i as za 2 
t t sae g 
{{ it jaan bist if 
el aA ITVNATAOATATT ‘ 
° _s 


Fig. 373. Commercial Probability Forms. 


These two sheets, an arithmetical-probability form and a logarithmic-probability 


form, are marketed under these names by the Codex Book Company.—Per- 
mission of the Codex Book Co. 


forms with the probabilities projection of the dependent vari- 
able scale, are published with both projections of the inde- 
pendent variable scale.! 


1The true nature of the “dependent” and “independent” variables in ogives has 
already been discussed. (See Chapter XXIX.) The same considerations hold. of 
the ogive on probabilities paper, but by accident some of the publishers of probabil- 
ities paper have reversed the arrangement in their printed scales, thus inadvertently 
hastening the time when, in the author’s opinion, statistical practice will correct 
itself and plot x-frequencies, y-range. 


PROBABILITY CURVES Ae 


The probabilities scale, calibrated in percentages, will never 
reach either zero or one hundred per cent in either direction, 


SIZE OF FARMS 
. Percent Distribution of Farms of Specified Number of Acres or Greater 
Unitel States 
1890-1920 
(Source:- United States Census) 


© Shes o we er One O 
1920 Q e 2 iS z cn 2 ° ” ~ fo) 
& we Ye Me Le ; S S < $ 
eo eS 3 a 3s < 
& a 
Cy 
ad ~~ 0 Yay b © c) ° 
¢ > _ a roy ° 
eee 80S ber Seas si 
Lo) a 
i coy a iD © LR ° 
‘ ° ° . o o 
1890 8 © 9 Fa 2 0 
= 
99 
98 
95 
90 
80 
70 
60 
50 
40 
30 
20 
10 
5 
ve 
1 
° 


106¢ 
Over, 


Number of Acres 


Fig. 374. The Logarithmic Probabilities Projection in Use. 


Note the close approach to straight lines. Also the ease with which, median, 
quartiles, etc., and interquartile range are found. 


for the true normal curve is asymptote to the x-axis and its 
100% parallel. We can therefore never plot the two limits, 


yes mee CH AR 


" wee “ ‘e 


0 and 100%, on this paper. We can only come as close as we — 
wish toward these limits by extending the scale on into the 
/ COLLEGE SALARTES 
ican Colleges 1 


Universities 
nstitutions 


Salaries in Amer 
Including Public and Private 
United Stetes 


nd 
I 


o 
e 


on) 


* (Source:+ United States Bureau of Educa 


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Gslary Ses Been Sele Sen Gare CR SMee Stans eae mee g 
iy bone eet sre tase a SSS a 


THIN 


Fig. 375. An Arithmetical Projection of the Dependent 
(or Frequency) Scale. 


small fractions near each limit. After we pass the points of 
01% and 99.99% the scale resembles a logarithmic projection 
so closely that we can add to it by logarithmic scales. But 
the tails of a curve are least significant, and it is rarely worth 
while to make this extension. Unless we deal with large series 
containing over ten thousand items and so grouped that the 
terminal groups have only one item each, we shall not need to 
plot points less than .01% or greater than 99.99%. | Usually 


PROBABILITY CURVES 461 


it is safe to chop off all of the scale beyond 1% and 99%, thus 
reducing the chart, either because of the absence of data out- 


_ COLLEGE SALARIES 
Salaries tn American Colleges and Universities 
Including Public and Private Institutions 
United States 
1920 
(Source:= Onited States Bureau of Educetion) 


© 
S 9 
Be Bd he el ee 
Presidente us ra wn ay oe te a 
3 o 8 eo S$ 8 
See eers ss 
Boans ond Directond $ ee Ky ss oe ae 
3 as Ss 8 8 63 
2s @ © o 
s 8 s ie o 2 
Pull Profestors 8 a a e 3°45 eo 4 © By” os eo °¢ 
Bie Onkose cua Supiee! (8 6S! 3. Oe 
2 
2 ¢ ° 
Associate Professors g Sai ee alas = fo 
& a & a 8 S 
5) 
2 
3 2g 3 a le s 
Aasistent Professcrs x Pee et le ey a Garth ae land 
$ S 8 S & 8 > ee % 
o god4 ° 2 2 
Instructor 8 S ao ee Ts ta he! S 2 3 or hed 
8 & 2 8 a & 8 3 3 
Avsiotents ee 4 ie ze * 3 % 8 8 g g 3 <) 
ac 2 3 2 8 = 3 8 nd 
x rts ek ae OSS Nn a ee re Aare 
Sale 7) ry 2° ° ° ° ° 
Woes ee Sa Sone ey No angi. 8) BSR) Bons 
eee en eS Ee ne ener S. a8 
SU | | 
98.9 F— 
e9 64 178 625 1010 640 1778 1040 
20 150 760 1160 1750 1250 1450 2200 
BO |_| 230 1050 1600 2000 1600 1750 2509 
300 1120 1670 2100 1720 1900 2800 
0 $70 1200 1640 2200 1850 2100 3000 
60 600 1340 1750 2350 2100 2360 3450 
50 I] 630 1600 1900 2500 2400 2700 3900 


740 1570 2040 2620 2670 $000 4500 


40 


800 1700 2200 2760 3080 *400 5200 
1000 1750 2300 2850 3300 3650 5600 
1080 1800 2400 2950 3550 4000 6000 


S00) ==7 


20 [TF 


1360 2000 2650 3200 4250 4800 6800 


io 


Percent of Persors Receiving Salerios 


2000 2600 3400 4000 6000 7750 9200 
: 
2g eens 4 
TacStar blue i i 
IMME nals eS Sameer’ Ste 
CPR cart. at ieet ies 
Se Orgs Salstye Sure sane 
a Seas peshas Segacrea tans 
28 Ski eebeck aa 


6G 


ies | ee L THIN | 
“at 8 i es ee ee ee 


So o 


100 


Dollars of Selary 


Fig. 376. A Probabilities Projection of the Previous Chart. 
Note the secondary data and bar-chart obtained from interpolation of the curve 
at the various decils, showing the median and other salaries for each group of 
educators. The swing of the upper tails of the ogives away from the straight line 
of the normal distribution might be interpreted as due to a few institutions 
which give titles out of proportion to the salaries attached thereto. 


462 CHARTS AND GRAPHS 


side this range (other than the limits) or because of the lack 
of their significance. ; 


The significance of the ogive is always elusive to the layman. 


and the probabilities projection of the ogive, while it clears 
up, for the technician, the question of normality, is even more 
baffling in other respects. A little study will show us, how- 
ever, that both the direction of the curve and its position are 
significant. When comparing two ogives, if we find the ogive 
of one series further out along the horizontal scale than the 


OUTPUT OF FACTORIES 


The Value of Products of Manufacturing Establishments 
shaving less than specified value of products 
per establishment 
United States 
1904-1914 
In percentages of the total 
Source:--U S Census 


J 
eae 
I 


RR WREAASE OS: 
225222 => 


Se ES ae eae I a edd Wad em 
——s 


YP wOnNaAaArR RIO 
° NOKTOHOMHORAAD 


a 


Senos 
Shen oe oe ee 
=== 

= 


) 
i 


rv 
° 


a 


nN 


Percent of aggregate Value of Products 


a 
5 
i 
J 
u4 
: 


| 


50,000 
100,000 
200,000 
500,000 
000,000 

2,000,000 


55 


000 
10,000,000 -—t—— 


Dollars Value of Prpduct per Establishment 
Fig. 377. Symmetrical so Far as Data Obtains. 


other, it means that the items in that series are greater than 
corresponding items (in similar parts of the distribution) in 
the other series. Thus an ogive for the heights of children 


PROBABILITY CURVES 463 


will be to the left of an ogive for the heights of adults. If the 
two simple frequency curves had-been plotted they might 
overlap but the main bodies of each would be at different 
positions along the scale. The significance of the positions of 
ogives will be clear if the corresponding simple curves be 


WHOLESALE PRICE CHANGES 
Distribution (by magnitude) of “Chain Relatives” of Wholesale Prices of 230 Commodities 
United States 
1891-1913 
(Total nunber of chain relatives showing change of price, 4,881; 697 cases of no change being omitted) 
(Arranged from Mitchell: "Index Numbers of Wholesale Prices”) 
(ean = 1,5% Incranse; Stardard Deviaticn * 14.44%; dotted line shows fitted normal) 


Persextaer of times a i a x eS aA eu e@ eran 

prices were greater 3 3 S x ir g Soe Se See ee See Cee era 3 
than specified per- 6 = a < Whee es TG: Meaty eo). a, aucens Gl eden 3 
centage of onange a ~~ GF F DHMH D BHAS 


99,9 
99.8 


: 
: 
iicce es OS AT 
U 
+—}- 
iWragli 


a 
ro} 


°o 


Percent of Frequencies 
Nn ua > a 
a. 9 ray 
| + 
{ at 
| [ 


10 et ert ine — 
e 
5 ——t i a +—t 
/ 
2 els +1 LL 4 { 
i; ein oo ie 
1 ol # ae 4 | ae es 
6 ma a eee! as + 4 | {|_| 
i | | 
a2 Z 
z al ial ro 
“S 3 3 §€§ 2° 2FRS2BsLSSILLRL2 


0 
6 
} ’ Percent of Change 


Fig. 378. Symmetrical but Distinctly not Normal. 


imagined underneath them or if horizonal bars be imagined 
as lying between each ogive and the y-axis. 

As in other ogive-charts, secondary data may be derived 
from the curves, being the reading upon the x-axis scale of 
intersepts of the curve and the horizontal rulings. In other 
words, while the curve is drawn by plotting the ordinates at 
certain abscissae we may interpolate from it the abscissae of 
certain ordinates. The readings are usually taken at the 


464 CHARTS AND GRAPHS ; 


median and quartiles, and decils, and occasionally at the ex- 
treme percentiles.2_ In this derived or secondary data we see a 
reversal of the dependence of the variables, the dependent one 


DURATION OF STRIKES 
Gumder of Strikes ended in Specified Periods of Time or Lest 
United States 
19161921 ° 
(Souroe:- Monthly Labor Review) 


Yeare Totale 


1921 1,409 & 3 8 §S8sRe F823 s3Snsh es 548 5 
192 ‘ Bice tets, Cause eitsueireace Bete d 
e A AARRK SVS FSSses sexe: 2 
iezo 1,600 S$ 8 $3384 eS &Ssees ss ess < 
— © S 82 SH8hS 299 Foes se sss 2 
1030 30088 3 SS S&S SSER Pee Sessse se ges 3 
—-- Go 7 A ASSSS FSF BLISS SE SEB 3 
rie 2s 8 2 § 28835 888 2agngge2 253 E 
par: 3 S 28 232385 8Se& Pessss ss ees 2 
wir 2.016 | $ 8 SSxseg sea gsrses 3s sss s 
=o 8 8 8 283988 382 ResssssgRz2e8 & 
we _2s8 § 2 ¢ ¢848e 548 sgasze ex gge g 
Sed be A € 84922 883 RP&&sss SSaas a 
99; : 
98 fT I Z 
L PALE 
Des ie ed 1916 = «19171918 1919 geo 
9 +4 My _ 
B AZ 
80} F 63 “ 39 86 108 126 
ia [ B »> a 
7 ¢ 
65) 5 7 
3 %-————_ ia if 26 24 22 $3 60 82 
os 1 17.6 18 43 47 68 
g %e}—— | ee 16 rt 14.6 360037 88 
S 60, 2 a un ° Ors) 26: 88 
‘pe Z Z 
60) Apt 6 6.2 0.6 18 16 26 
1 
Car t $6). 4.0, 4.37 23 10 18 
pS a 1 
30) ate “me | 40 28-20) 48 to 8 
26) ROS ah Tt | SY 2, Sh RL 6 CF. 
a = 2.4 1.0 1.6 408 3.9 6.4 
ler Z 
19> + 1.7 Zl 18k 
ae 
ae | 
6 L 
tol Pe Ee aS Nee SOULE Ss & 
Woeks:< Cd . no #neor aang 200 
Wonthe:> - - 


Qurotion of Strikes a 


20) 


Days of Strikes 


pt 
1916 191T 1918 11g 1920 iveh 
Yeara 


Fig. 379. The Comparison of Ogives for Different Dates. 


This gives a historical curve instead of a bar-chart for the secondary derived 
data. The ogives are useful for interpolation but the derived curves afford the 
best view of changes. 


* That the interpolation for median, quartiles, etc., does not give cases, but values, 
has already been pointed out. (See Chapter XXIX.) 


PROBABILITY CURVES 465 


becoming independent. This secondary data affords most of 
the statistical measures of both dispersion and skew. 

The significance of the direction of the ogive upon the 
probabilities paper (when it approximates a straight line) is 
more subtle, but far-reaching. For as we have seen, the range 
of observations may vary in different series. In the example 
already given of measurements of a given distance, we saw 
that as precision of measurement is increased, the distribution 
becomes more concentrated, the scatteration less, and the peak 
of the simple curve taller. It is precisely this condition which 
will make the ogive curve more nearly perpendicular to the 
x-axis. As the dispersion increases and the precision dimin- 
ishes, the ogive curve swings about toward a horizontal direc- 
tion. These considerations hold as well for the ordinary ogive, 
but are more clearly seen and measurable in the ogive pro- 
jected upon probabilities paper.® 

In the analysis of frequency data by means of the ogive 
curve upon the probabilities projection, a difficulty is often 
met in that the data is incomplete, no figures being obtainable 
for a portion of the range. Thus the statistics of income do 
not include the personal incomes below two thousand dollars 
for heads of families and one thousand dollars for individuals, 
and for this reason omit perhaps ninety per cent of the popu- 
lation. Astronomers have estimated the number of stars of 
each magnitude down to the twentieth, but do not carry their 
estimates much further. In such cases as these we do not 
know the entire “population,” “universe,” or total body for 
which the distribution applies, and hence cannot turn fre- 
quencies into percentages. 

The problem is largely statistical but affects the subject 
of charts in that an ogive upon probabilities paper, of these 
incomplete distributions, obtained by turning the frequencies 
into percentages of the known sub-total, will almost certainly 
fail to form a straight line although the entire distribution 
may be perfectly normal. It is not legitimate to plot such 
parts of distributions upon the probabilities paper unless we 
can turn the frequencies into percentages of the true total. 
At this point, therefore, we will mention a statistical trick by 
which you can sometimes dodge the difficulty. For in all fre- 
quency data there are two possible series, both covering the 


On the probabilities projection, the mode can not be graphically determined by 
the slope of the curve, as it could on arithmetical projections of the ogive. 


Sfsk-LIGHT 


Relative Number of Stars of Various Naznitudes and Quantity of Light Therefrom 


(More-than cumulatives only. Full lines for plota as per scalos shown, dotted 
lines for arithmetical projection of scale for relative brilliancy.) 


Light 


Stare 


99.9 
99.8 


99.6 
99. 


90. 


96. 


90. 


85. 
80. 


70. 
60. 
60. 
40. 
30. 
20. 
"18, 
10, 


6. 


2. 


1. 
0.6 


0.2 

0.1 

0.08 
0.02 
0.01 
0.005 
0,002 
0,001 
0.000,5 
0.000,2 
0,000,1 
0,000 ,05 
©.000,02 
.000,01 
0,000, 005 


©.000,002 


0,000,001 — 


Fig. 380. 


0 
oa 5 A & wo on 
oOo 6G Ht Hh FH HH 8 oa o nwo nm 30 Owe EF FE ew 
GaN) Rt He eet Exe RT OTe Ae Ea eee Sees at Ay Aer IS. Gon) Soar 
Go O° we ey OO) ee a * Ties ep ye Be et 20) a i Cece ed 
8S 88 FS 83 oF 2 8 BS nN GA 
a Oe 
un 
5. SN OST SS. | Shae ae ORS 
wo a 9 
wo e334 $5 8 8 88 
aA a nH om + AS 9: Oi nd QO: 69S 
© et ed Oe ta lS SE) Bi ey 3S Sa 1S) SS ta 8 
Oo rw AOS AE POO Te OS Pes ee oe TED! Cy SP ES LOY EO Oe OS OS 
Cant ae em SPS a Er RN rot * rte os eae re os cae es cy yal 
o.-¢ & + 0) 6) >. 
eC oe Sg ca 


ie 
al 


| 
| 


ne i hee ‘ if | 
— a > 
se — i + 1 +——_}— + 
he | i { \ | 
: | | i 4 Tae al 


a 
\ 
s 
ae 
. 
a + 
ae 
eS oo Dn © 
or 8 + Sunn 
8 2 mh biet) Lier Red, RE SOROS SLOTS Te Say eS Meats 
q Magnitude 
a 6e© F&F @® 
ee it 
= © oa 
i) wD Vee vowel to 
SS Ae MS Oe | eres Lon) 
. 8g 88 Sk 3S een, 
38 ° 7 pee eo) an 
Se ee ES Ge Se & g 
AOR eRe Ba 8 8 6 ne Bie ee ea eae ete 
Relative Brilliancy Pa: ate e 


The Ogive of the Units of Measurement of the Items 


Brilliancy) in an Incomplete Series is Straighter and more 
able than the Ogive of the Items (Number of Stars). 


See footnote on opposite page. 


(Star 
Reli- 


PROBABILITY CURVES 467 


LABOR TURNOVER 
Separated Employees and Equivalent jiunber of Fulleyexr Jobs, Subjoct to Instability 
FE classified as to length of service. 
F x ( Employees * percent distribution of 2,581 separated employees) 
("Jobs" © percent distribution of aggregate length of tine served by 2,563 separated enployses 
who served less than five years) 
Sugar Refinery 
California 
Year Ending way 31, 1916 
(Seurce:+ Paul F, Brissendoa’ 


Jobs 99.0 97.3 92.6 75.7 6.4 43a 26.7 19.3 0.0 


Enployees 78. 65. 46. iin 12, 6. 3. 2. 1. 


za mess 
vd 


Li 
ite 1 3 6 nt 2 3 5 
(-- Jeeks --) (--------- - Months ------------ ) (---------- Years ------- =--) 


Fig. 381. Another Example of the Two Interconvertible Frequencies 
for the Same Data. 


e 
same observations. These two series have been described in 
an earlier chapter. They are, briefly, the count of items, and 
the count of units of measurement of the items, group by 
group, through the distribution. The point is that these two 
series for the same phenomenon are often available and can 
usually be estimated if not available. And if a considerable 
portion of the data of one series be missing, it is generally 
true that the missing portion covered items of small or negli- 
gible numbers of units individually. If then we convert the 
Note To Fic. 380 
The dotted lines are plotted upon an arithmetical scale (not shown on the chart) 
of rélative brilliancy. The full lines are plotted by the two horizontal scales 
(appearing on the chart), namely a logarithmic projection of relative brilliancy 
and an equivalent arithmetical projection of magnitudes. It would also have 
been possible to project the magnitudes logarithmically (thus obtaining a log-log 
projection of brilliancies). The interesting point is that star magnitudes though 
projected arithmetically are in themselves geometric units and form a logarithmic 
projection of brilliancies, 


468 CHARTS AND GRAPHS 


item data into unit data, we generally find that the importance 
of the missing portion of the data has greatly diminished. Thus 
while income statistics omit perhaps ninety per cent of the 
families and individuals in the country, they omit only about 
ten per cent of the total income. In the case of stars, the 
astronomers have computed the light of all stars, so that the 
unit data is complete, while the item-data is incomplete. 
When the incomplete data can be reduced to a relatively small 
amount in unit-data, it may more safely be estimated, and 
so completed. Thus by converting the data into another 
form, we may find it possible to project the ogive curve upon 
probabilities paper with satisfactory results.‘ 


OUTPUT OF FACTORIES 


The Value of Products of groups of Employees 
employed in Manufacturing Establishments 
(all groups composed of employees in establishments 
having lowest value of products) 

United States 

1904 - 1914 
(In percentages of total) 
Source: U 8 Census 


Establishment 
by Value Employees| Product |Employees |Product |Employees|} Product 
of Products Number Value Number | Value Number 


Less-than $5,000 1.9 202 
Less than $20,000 9.6 


Less than $100,000 2804 


Less than $1,000,000 1404 


Any value whatever 100.0 


Fig. 382. Alternative Data Yielding the Lorenz Curves. 


4 “Tf the observations are not complete (i.e. cover only a small part of the unknown 
total range of the variable, though it is highly desirable that what observations we 
have do not constitute a mere extreme tail), it is possible to fit a‘normal curve to 
the data by means of fitting a second degree parabola, by the method of least squares 
(y=4+Bx+Cx*) to the logarithms of the number of observations in each interval. 
The theory is based upon the equation of the normal curve, 

=o 

: ; Bees .¢ae 

Taking logs of both sides, y= Kye 


Berar 
loge y=loge K,— — x7 


=K,— Kure 


we get a second-degree parabola.’’—Dr. Frederick R. Macauly. 


PROBABILITY CURVES 469 


We have spoken of the alternative series into which any 
distribution may be converted. The combination of these two 
yields, upon arithmetical paper, the Lorenz curve. And it is 
for the Lorenz curve, of all types, that we can profitably use 
double-probabilities paper, that is, paper projected upon the 
probabilities scale along both axes. For, as will be remem- 
bered, the tails of a Lorenz curve are nearly asymptote to the 
axes when the dispersion is great, and the values near either 
extreme become difficult to interpolate or read from the chart. 
Moreover, when the groups or classes, into which the distri- 
bution has been arranged, are few, the curvature of the curve 
becomes angular and interpolation is unreliable throughout 


QUTFUT OF FACTORIES 


The Value of Products of groups of Emplcyees 
employed in Manufacturing Estublishments 
(a1). groups composed of emplcyees in establishments 
having lowest value of products) 
United States 
sisiwenwoe eee Gt 
—— ——1909 
1914 
Source; U S Census 
(In percentages of the total) 


ao 
Oo 


=] 
oO 


Qa 
fo) 


oC 
S 


Pe oO 
fo} 


a 
fo} 


se) 
[o) 


v 
°o 


Percent of aggregate Value of Products 
o 


7 


we 5 . 10 20 30 4050 60 70 80 90 
Percentage of aggregate Number of Employees 


Fig. 383. The Double-Probabilities Projection Straightens Out the Lor- 
enz Curves When of Normal Distributions. 


the length of the curve. But when both axes are ruled on the 
probabilities scale, the tails become indefinitely long, the zero 


PERSONAL INCOMES AND TAXES 
Distribution of Income and tax emong tax-payers 
United States 


, 1919 
(Source:- Colfector of Internal Revenue) 


INCOME CLASS CUMULATIVE (Less-than) PERCENTAGES 


Over 4 
Income Tax 
gre oal cone, te Returna (eet) (total) 


Under $ 2,000 36.09 
a 65.53 
87.67 

95.90 

98.94 

99,64 

99.89 

99.95 
99,986 
$9,995 

99,999 

100,00 


Fercentage 


w e oO @ 
e e 
Percentage OOS a” C2 G2 


tribution: 


PROBABILITY CURVES 471 


and hundred per cent points disappear, receding to infinite 
distances, and the curve, throughout its length, becomes very 
close to a straight line. When the distribution is normal, ob- 
viously the curve becomes a straight line, and hence the 
straight-line Lorenz curve on double-probabilities paper is a 
quick and useful indication that the distribution is normal. | 
Obviously the accuracy of interpolation is improved by this 
straightening out of the Lorenz curve, as is also the facility 
for detailed comparison, such as through light-analysis, of 
several distributions so plotted. 

The utility of the probabilities projection must be ap- 
parent to those who deal with frequency data. For analytical 
purposes, as a labor-saving and illuminating chart, it takes its 
place beside rate-of-change paper for historical data. 


CuapTer XLI 
SHIFTED ZERO-POINTS 


We have seen, thus far, three great types or kinds of scale 
projections, arithmetical or uniform, logarithmic or geome- 
trical, and normal or probabilities. We have seen these com- 
bined in every way upon the two axes of the chart. All 
this has been done in the search for simplicity or regularity of 
behavior, and convenience or ease in interpolation. There 
remain still other projections of the chart-scales, which serve 
the same purposes and will be discussed in later chapters. 
And there are a few minor variations of the logarithmic pro- 
jection which can well be discussed here. 

We have seen that historical data can be plotted with either 
logarithmic or arithmetical vertical scales, but that it is not 
correct to use anything except an arithmetical scale on the 
horizontal axis. ‘lo this rule we may now note two exceptions. 
The first arises in the case of data with a definite crigin point. 
Thus the pseudo-historical frequency series which involve time 
have already been put upon logarithmic x-axis scales together 
with other frequency series. From these frequency series in 
which time is the independent variable, it is but a short step to 
strictly historical series, in fact the distinction disappears here, 
the same series being called equally well a frequency or a 
historical one. But there is also a class of purely historical 
data, involving specific points of time, which can ke placed 
upon a logarithmic x-axis. This is data, generally of a geo- 
logical, or other scientific nature, covering very large periods of 
time, such as the age of the earth, and its important geological 
eras. 

The second variation of historical series is extremely 
interesting, though of very limited application. It may be 
called the retrospective projection. If from any point of time 
we look backward over the years, we may notice that the more 
recent events stand out more clearly, and in more detail, while 


472 


SHIFTED ZERO-POINTS 473 


the events of early years become more vague and their details 
lose importance. In business, this importance of the last years 
is recognized and business statistics therefore often contain full 
detail for the most recent period and only brief summaries of 
previous periods. In histories, the space devoted to ancient, 
medieval, and modern times, usually shows a similar com- 
pression of earlier times. If these witnesses are of any value, 
they testify that the importance of detailed data diminishes as 
its remoteness in point of time increases. And the chart-maker 
has therefore a legitimate object in devising a chart method to 
display the data in its proper detail or lack of detail. 

Several methods have been tried to meet this charting need. 
By the silhouette bars a few facts of the past history are given 
in addition to the very latest figure. By the juxtaposition of 
two curves with a single y-axis scale but different x-axis scales, 
one, let us say, for years, the other for months, data for a recent 
period can be given in full detail, and that of a previous or the 
entire period in summarized form. But the inventive mind 
will seek still a better method, which will not have the rigidity 
of the last and in which the disappearance of detail will be 
gradual and so we arrive at the use of a logarithmic x-axis scale 
projection, reversed in its direction so as to compress the earlier 
periods of time upon the chart. 

This retrospective logarithmic projection of the time scale 
has both intriguing advantages and baffling disadvantages. 
Upon its credit side we may observe that it presents precisely 
the degree of importance to data at various points along the 
line that we desired and has unlimited possibilities of extension 
backwards into remote antiquity without consuming space 
wastefully. If we are, in the year 1900, let us say, to look back 
over the centuries, we shall doubtless attach the same relative 
importance to the entire nineteenth century as we do to the 
seventeenth and eighteenth combined, the same relative impor- 
tance to the last thousand years as to the two previous thousand 
years. Important exceptions occur, of course, but in the main, 
this proportion of weight of importance holds, else historians 
would not be justified in devoting their space to the different 
periods in these ratios. The real nature of that phenomenon 
of change which we call the passage of time is still a profound 
mystery and it is an interesting speculation that it may in some 
occult way combine elements of progressive and regressive 
organic growth, But idle as this thought may be, the chart of 


474 CHARTS AND GRAPHS 


OsterLoeys IH THR WORLD 


Thanpleyed Percentages among Trade-| Union members in the chief coutrise 
1915-1921 
(Soaroes= Monthly Labor Review) 


@ 6 <s = a < 
Labag tiga hy Claes) as Ki) 3 Ss ¢ 3 2 
os wo 
sentimvia 3 gf 2 S} ; 5 i. Gacadade as «8 <5 5 
rea Ate ant at 
Canada 2S = 3 = Sau aes ata roma Seine 
Gereany ef eRe hd & 3 “ = SSS rate Se Gugiciea: Sane 
” 2 le 
Gret Britain J of 22S ¢ 3 z noeclrte ectied alge ah pearl Pens oie 
[ 3 
80 == 3a 
eC : = 
| ital 
| 
2 = 5 si : 
ioe 
¢ f f 
is t ; 
4 
3, 
P| 
| 
Bol Pro [ i 
oe I : 
9.0 : = 
O4 a2 - 2 2 
“aver Aver = Avor = Aver Aver Aver Aver g2ahyhar 8s bed Fa 
y1e 1919 
1913 1914 «19181916 1917 4920 ase 


Fig. 385. A Historical Retrospect with Reversed Log Plotting for the 
Horizontal or Time Axis. 


The purpose is to present recent developments in greater detail. 


the geometrically retrospective historical curve has a certain 
value in presenting graphically a survey of the past in proper 
emphasis and detail, and affording a comprehensive picture not 
otherwise equalled. Owing to the fact that the significance of 
the slopes of the curve disappears in this chart, its usefulness in 
mathematical curve-analysis will always be limited, if not 
doubtful. But when the gun-shot method of plotting be 
employed (that is, isolated points be plotted) its success 1s 
marked. It is not improbable that in time all school histories 
will be illustrated with diagrams in which events will be entered 
at their proper positions upon such a scale. 

The chief disadvantage of the logarithmic retrospect chart 
lies in the fact that before entering the time-figures upon the 
scale, these figures must be computed back from some origin 
point, either in the present or in the future. The choice of 
origin-point for our backward count of the months or years 
directly effects the degree of expansion which the most recent 
periods of time will undergo on the chart-scale. If we take the 


SHIFTED ZERO-POINTS 475 


WHOLESALE! PRICES IM THE WORLD 


Index mmbers for the chief nations 
1913-1921 
(Source:~ Monthly Labor xeview) 


sag nian aes & 8a 
Gmy 8 & § g 2 Sess ae ses Skee) Sees 
8 = °&8 2 Supe ee BS es oe eS 
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a 2 duraa 2 oe ercodwan wor e on oO a o < 2 a 
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United MQaxArA Oe D AN Oo eA YO oro At © 9 9 O22 aang Oo S ry i c 4 
Sate oo 8 @SSSSsaRggkss SSssegssgs FF F #8885888 R8 &§ ZB oe 
- az) iS -A- 1 “| aid BS as df =} 
| aps 
900 a = \ge 
600 
700 
600 + 
i 
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al 
400- r 
800'—+ te 
I 
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Jal 


0 eau ret ae ae 
93 4883983 
21s wm 1908 1910 


Fig. 386. Another Example of the Same. 


immediate present, then last month’s figures will be plotted at 
unity, as being one month back, the previous month’s data will 
be entered at the logarithm of two, on the left side of unity, and 
so on backward along the axis of the chart. But where will we 
plot next month’s figures when they appear? We cannot plot 
them at the logarithm of zero. In other words such a retro- 
spective chart would have to be redrawn each month. The best 
way to avoid this is to assume an origin point of time some 
distance in the future. Although this destroys the significant 
relation of the more recent events, so far as that relation was 
desirable, yet it enables us to bring the chart up to date for 
some time to come. But the student will now see that com- 
parisons cannot be made from one such chart to another unless 
common ‘‘view-points” have been used in both. The dis- 
advantage is so serious as to make the chart useless for research 
purposes, but it is still worthy of notice for general records and 
popular presentation.! 


1For short periods of time, a “squares” or other powers projection will serve 
equally well and can be continually added to. 


476 CHARTS AND GRAPHS 


- The use of altered logarithmic scales for the frequency series 
is a step which requires much more technical mathematical jus-_ 
tification. We seek here to bring not detail nor convenience, 
but symmetry and regularity, to the curve. Experiment will 
show that a great many asymmetrical distributions can be 
made to approach symmetry by assuming false origins and 
correspondingly altering the logarithmic scale projection. That 
this is necessary for data in which the original zero is a false or 
arbitrary one, having a real positive value, has already been 
pointed out. But when cause for shifting the zero does not 
clearly exist in the very nature of the data, the student should 
be slow to alter it, for even the most excellent symmetry which 
may be induced thereby may be utterly lacking 1 in significance. 
This applies not only to shifting of zero points, but also, of 
course, to reversing of directions from the zero, a step which is 
closely allied in that it amounts to giving to some value above 
the maximum of the range an assumed value of zero and treat- 
ing the resulting negative values in the range as if they were 
positive. 

There is open to us another alteration of the x-axis scale - 
projection which is not of any value for historical curves, 
namely, the anti-logarithmic projection. For just as we have 
been able to plot the logarithms of our scale-figures, so, too, we 
can plot their anti- logarithms. The device has resales similar 
to the retrospective projection, in that it expands the larger 
numbers and compresses the smaller ones. Here, too, caution 
must be used in attaching significance to the results. But the 
mathematical interpretation of this projection, though still 
technical, is much simpler. There are in fact certain classes of 
measuring units which are definitely of a geometric nature and 
not arithmetical. It would be useless to plot these upon 
logarithmically projected scales, for the numbers themselves are 
already logarithms, and the log scale would really give a log- 
log projection. ‘Thus stars are classified as of various mag- 
nitudes, each magnitude being two and a half times as bright 
as the next. Musical pitch is measured in tones and octaves, 
each octave having twice the wave-frequency of the preceding 
octave. In these cases if we seek an alternative projection we 
must obviously use the anti-logarithms of the magnitudes and 
tones. Both the anti-log and the log-log projections may 
occasionally be useful upon either axis of the chart, 


CuapTer XLII 
CURVE-FITTING 


The reader has now seen a variety of ways by which sym- 
metry or regularity can be brought to the curves of historical 
and frequency data. The reason, as he has seen, behind this 
quest for simplicity, for straight lines, for parallel or mirroring 
curves, and the like, lies in the ease with which generalization 
can proceed from such forms, and the degree of confidence with 
which we may accept the data as reliable samplings and their 
curves as significant pictures. When we see the curve of the 
country’s population mounting higher so steadily that it forms 
one single straight line, we know, without performing any 
mathematical exercises at all, that the population grows at a 
constant rate, and we are thankful to the logarithmic projection 
of the chart which has yielded this simplicity. When we see 
the curve of our advertising appropriation paralleling, on rate- 
of-change paper, the curve of our gross sales, we know without 
any computing that the same fraction of the dollar has gone 
into advertising every year. When we see the cumulated curve 
of the nation’s population as divided into cities of various sizes, 
forming a straight line upon the probabilities paper, we know 
that without appreciable error or effort we can by interpolation 
find the number of persons inhabiting communities of any 
particular size, whether or not the Census has mentioned com- 
munities of such size. And in every case, the same causes 
assure us some degree of confidence in the reliability of our 
observations as fair samplings, when we have put partial data. 

The question of the reliability of data is properly a statis- 
tical one, for which the student should consult the statistical 
authorities.! It arises in the collection of data, before charting 
has begun, and only recurs again when the near approach of a 
charted curve toward regularity and simplicity raises the sug- 


1 See particularly Bowley, p. 178. 
477 


478 CHARTS AND GRAPHS 


gestion that the deviations of the actual curve-line from the 
desired simplicity of form are due to errors of data. Thus if 
the ogive of the distribution of incomes is very near to a straight 
line on logarithmic paper, it is but natural that the theory 
should arise, as at least a tentative explanation, that the 
deviations are due to omissions in tax collection, evasion in 
income reporting, or in some cases chance variations due to few 
observations. 

And it often happens that in the desire to justify a theo- 
retical simplicity, we are too ready to excuse deviations from 
it as errors in the data or chance variations. A straight line 
may fit so closely to the data that we feel sure that it, instead 
of the observed curve, represents the truth. Thus the entirety 
of Pareto’s law of incomes is a result of adopting the fitted 
straight-line rather than the actual income curve (ogive). 
Since the income curve is truncated at its lower end owing to 
lack of information on incomes below the tax limits, that law 1s 
based upon insufficient data. And recent investigations lead 
to the belief that the true ogive of incomes is not a straight line 
upon logarithmic paper, but upon logarithmic probabilities 
paper.2 Slight deviations from a straight line are not con- 
clusive evidence of errors in the data and the student should be 
extremely careful in drawing hasty conclusions from a close 
approximation to a straight line upon the special projections 
which have been described. 

There is, therefore, always a danger that the close approxi- 
mation of a given curve to a straight-line, or other simple 
theoretical curve, is fallacious and deceptive. This caution 
cannot be too strongly emphasized. It attaches prima facie 
to all attempts to fit theoretical curves to actual ones and 
casts upon him who would fit such curves the burden of proof. 
The presumption is that the deviations of the given curve, 
from the fitted one, are significant. The removal of this 
presumption may call for all the analytical powers of the 
statistician, but we should always start with the presumption, 
and never lightly abandon it. 

There may be many reasons why the deviations are insig- 
nificant. Some of these may be found in the particular cir- 
cumstances surrounding the collection of the data, such as 
bias on the part of the investigators, or difficulties of observa- 


2See National Bureau of Economic Research, Income in the United States. 


CURVE-FITTING 479 


tion. But, whatever else may be found, there is likely always 
to be one cause for the lack of a perfect fit, in what are com- 
monly called “chance variations.” These are more marked 
in small samplings than in large ones. A definite mathema- 
tical law for the probability of this occurrence can be found in 
books on the subject. The reader who recalls the normal 
curve of error will understand the inevitability of such chance 
variations. And, needless to say, when we can safely consider 
the deviations of a given curve from a fitted theoretical one to 
be due to chance variations, these deviations lose all signi- 
ficance and we may safely proceed with the fitting. 

The theoretical curve which we propose to fit to a given 
curve may have any shape. The simple linear curve or 
straight- line upon plain paper with arithmetically-projected 
scales, is merely the simplest of these. And the reader will 
remember that, in the discussion of cycles in historical data, the 
fitted straight-line was called the “secular trend.”’ It is a very 
crude secular trend, convenient, but in most cases not pre- 
cise. The reader has since seen that, for most economic 
data, a straight-line upon the rate-of-change (or semi-log) 
paper would be more accurate. Still other “trends” and 
straight-lines will be described in later chapters. For frequency 
data, the fitted curve is generally the normal curve, or its 
equivalent straight-line upon probabilities projections. Of 
the significance and appropriateness of these straight-lines, 
in each case, the reader has already a general understanding, 
and the mathematics of these and other straight-lines will be 
discussed later. | 

While so much attention is being given to the various 
charting methods by which theoretical curves are reduced to 
straight lines and by which actual curves are more easily com- 
pared with theoretical ones, it would seem well to mention 
briefly the mechanics of fitting. This problem arises after 
the particular theoretical curve to be fitted has been chosen. 
Let us assume that it is a straight line upon one of the chart- 
forms already described, such as the semi-logarithmic or rate- 
of-change paper, or the probabilities paper, or even the plain 
uniform paper with arithmetically projected scales. The 
problem of fitting is virtually the same in all cases. 

Whenever a straight-line (in the example we have taken) 
is to be fitted to a given curve, and that curve does not form 
in itself a perfectly straight line, it is obvious that the straight 


480 CHARTS AND GRAPHS 


line may lie in an infinite number of slightly different positions 
and still fit very closely to the given line. The problem is, 
therefore, to find the particular straight line (or other theo- 
retical curve of the selected type) which gives the best of all the 
possible fits. If we call the deviations of the given curve from 
the fitted one, its ‘‘residuals,” the problem is, broadly speaking, 
to find the fitted curve which makes the total of these residuals 
a minimum (that is, the least possible sum for the given curve). 

There are three outstanding methods which have been 
developed for determining the best fitted straight-line. These 
may be called the graphical method of selected points, the 
method of averages, and the method of least squares4 Of 
these, the first is the simplest; the last, the most accurate; 
and the second, the most satisfactory because both fairly simple 
and fairly accurate. The graphic method of selected points is 
nothing more than laying a transparent straight edge or tightly 
drawn piece of thread over the curve and adjusting its position 
until an equal number of points appear on both sides of the 
straight line and the fit appears optically most satisfactory. 
The other two methods are mathematical processes, for which 
the reader will have to consult the proper statistical authorities. 

But the first of these mathematical processes for deter- 
mining the position of the fitted straight line, namely the 
method of averages, is also capable of a graphic solution. If 
you join the first and second plotted points and plot a new 
point midway on their joining line, this new point will repre- 
sent their average. If you repeat the process with the third 
and fourth, the fifth and sixth, and so on with each successive 
pair, you can reduce the whole curve to a slightly shorter curve 
with only half the number of plotted points all of which are 
averages. On the new curve fresh averages can be plotted, 
this time representing averages of averages, or averages for 
four points on the original curve. After repeating this opera- 
tion a sufficient number of times, you can reduce the longest 
curve to a series of two average points, through which a fitted 
straight line can be projected. 


’ Because the algebraic sum of the differences from the mean is always zero, 
statisticians often use the squares of these deviations. Strictly speaking, therefore, 
the problem is to find the fitted curve which makes the total of the squares of the 
residuals a minimum. If the residuals alone, instead of their squares, be used, we 
must seck to make the arithmetical sum (that is, the sum of the residuals, disregarding 
their signs) a minimum. 

*Cf. Merriman’s Method of Least Squares or Bartlett’s Method of Least Squares. 


CURVE-FITTING 481 


For many purposes it is sufficient to fit curves by inspec- 
tion, just as it is sufficient to correlate them in this way. The 
graphic analysis, made more precise by “light analysis,” is a 
tremendous labor-saver, and may serve at least in the pre- 
liminary stage of the study, at least. 

In fact, curve-fitting is but a variation of correlation, 
being merely the determining of the theoretical curve which 
best fits or correlates with the given curve. And the con- 
siderations affecting correlation likewise govern curve-fitting. 
For precise purposes, the mathematical method of least squares 
should be employed and a mathematical coefficient of correla- 
tion and probable error be computed to measure the success 


of the fit. 


CuaptTrer XLIII 
SPECIALLY PROJECTED SCALES 


We are about to embark upon an orgy of distortions, 
modifications, and special projections of the scales for curve- 
charts, all of them being designed graphically to facilitate the 
study of particular data. The general principles and the 
more useful forms for the non-mathematical reader will be set 
forth in the present chapter. In the succeeding chapter, the 
mathematics of all special projections will be discussed, from 
which any particular projection can be designed. The present 
chapter will suffice for most. 

It will by this time have occurred to the reader that the 
scale figures or calibrations may be plotted or graduated at 
any points along the axis of the charts which we desire, and 
can therefore be made to express any function of the variable 
plotted thereon. Thus the logarithmic projection is merely 
one in which, if we can designate by X the scale-figures or 
calibrations and by x the actual distances at which these are 
placed along the axis, then in the logarithmic projection x = 
log X (and X =anti-log x and 10*=X). From this it is no 
difficulty to proceed to the scale projection of other functions 
of the variables. A very simple example of this would be 


the expression of reciprocals by the scale x -+ ( or inc). 


Obviously such a scale along an axis would straighten out all 
curves in which one variable varied with the reciprocal of the 
other, and if both axes be plotted on such scales, then curves 
would straighten out when the reciprocals of both variables 
vary together. 

Perhaps to the economist the most interesting application 
of special scales lies in a recently discovered use of what may 
be called a “‘square-root projection”’ for certain historical data. 
The speculation which leads to the use of this projection is 
founded upon the analogy of familiar physical laws governing 

432 


SPECIALLY PROFECTED SCALES —_483 


the intensity of light, the flight of falling bodies, and the like, 
in which one set of values varies (directly or inversely) as the 
square of another. In the case of light, as every one knows, 
its intensity varies inversely with the square of the distance 
from its source and the area of the cross section of a beam of 
light varies directly with the square of the distance. Falling 
bodies travel in each unit of time over a distance proportiona! 
to the square of the number of units of time they have been 
falling, and their velocity therefore varies with the square of 
the length of time elapsed since leaving a position of rest. The 
idea suggests itself that certain economic phenomena may 
closely parallel in their growth the growth of such natural 
phenomena. It would be obvious, of course, that this method 
of analysis could only be applied to phenomena which are free 
of elements of organic growth or other factors, or in which 
corrections can be made for such elements and factors.1 
Other powers and roots may equally well be the basis of 
the special projection. These have not, however, as yet be- 
come important to the economist; they are chietiy useful in 
engineering and the natural sciences, where formulae and 
equations are found of every type, and degree. The conic sec- 
tions, the circle, ellipse, hyperbola and parabola are all im- 
portant to the scientist, while the economist does not need 
to go beyond the parabola and the hyperbola when he leaves 
the straight-line. Indeed the statistician either in business 


1 The Gompertz curve, as it is sometimes called, to which much economic data 
fits, is not unlike the curve which straightens out on a square-root projection, when 
the curve has been plotted upon a logarithmic projection instead. In other words, 
the square-root projection suggests itself for all economic data in which the curve, 
like an ogive, seems to have a “die-away’”’ approach to a maximum when plotted on 
log-paper, as if reaching a saturation point. 

The Gompertz curve is, however, probably nearer to the typical behavior of 
economic phenomena through their initial stages, from discovery and through experi- 
mentation and installation, to the final stage of “saturation” in which maintenance 
and upkeep constitute the chief requisites. This curve has not yet been made the 
subject of a special projection. Its formula is 


y= abe”, or log y=log a+c* log d, or loglog SS =x log c+loglog b 


indicating that a loglog y-scale with shifted zeros and an arithmetical x-scale will 
straighten the curve, when the value of the constant a has been determined. For a 
recent excellent discussion of this curve and the methods for determining the constants, 
the reader should see Prescott, Raymond B., Law of Growth in Forecasting Demand, 
in the Journal of the American Statistical Association, December, 1922, pp. 471-479. 
See also Running, Theodore R., Empirical Formulas, John Wiley & Sons, New York, 
1917, pp. 29-33. : 


CHARTS AND GRAPHS 


484 


Werld, 1600-1919 
(Sourse:= U. &. Statistical Abstract) 


THE WORLD'S 
Botinated eammerce and commoroial equipment specified 


vessels 
) 


Sailing 
(#.000 tons) 
000 miles 


Telegraph 
(-,000 miles) 


100,000 


(+,000 miles) 


Cables 


Fig. 387. The Four Lower Curves Fail to Straighten Out 


on Logarithmic Vertical Scale. 


2.23 
= Ys 
oOo 
aad 
Le ges 
a & 
wEL 
20,0 
pees 
are 
eee 
"OG gy 
PS ee 
as s 
Bes 


h such 


it 
data as the scientist, and cannot so 


ly works w 


or economics rare 
mathematical generalization, in the s 


equations. 


SPECIALLY PROYECTED SCALES 485 


THE WORLD'S COMMERCIAL BOUIPMENT 
Setimated Railways, Steamships, Cables and. Telegraph 
World, 1820-1919 
(8ourcese Us. S. Statistical Abstract) 


wo 
N 
o 4 
e e 
Cables d SA Sigua oe 
(Thousand miles) a NN NM 
on wm ete OO Oh Fr OF 
4 wz @o a o v tad 
Steam Vessels % Am ® a a a a od 
(Thousand tons) a oe 8 ON @ 
o 0 Oo fF & 
Telegraph ickh © cs 2 2 9 5 
(Thousand miles) Ga Se epoca 
«7 Foal 
Railways ae ts a 
(Thousand miles) Siw aeRO 105.0 (94 bac 
Aw Mm Oo oO fF 
6000 


a 
Lo EI 
or een 7 


PCA 

Leese 
Bae ieee 
Felechecbhodes tee 


NA || 
ie 
es 
N 
SS 


SSS 

a 

TD SSS 

E+“ 

: ESB Sa ea aS 
SP eee SS 
EADY EG! BEA Si Ree 
Pg Re en 2 [il rere ae ae ae a | 
Aa ey i ta 
[TS EA eae J Ea 


rr Cea 
+ ie oa 
02900 0 0 owe 
@ @ 
~ - 


Fig. 388. The Square-root Projection of the Vertical Scale Brings 
Much Greater Regularity to the Curves of the Preceding Chart. 


486 i Projections of the Sino Curvé 


Amplitude 


(Any scale) 
| 
| 
| 
ities 


x)yy 
5 Cie soale) ‘> 
Periods 
100 
90 
80 
g H 
40, f 
@ 20 
= 6 
Sr=20: 
2 
— -+4 
Z 760 fait a0 ae 1 Ht is 
=80 | est 
290 oe 4 | 
x x 1 
2100 
(Any scele) 
Periods 
cae 4 
pa PS, SY ens 
® C ——+ +— 
pS Se 
roe 2a See | ic i Se 
© 
F —| dle Sey | aa 
Ly Es Sie 
as =e 
sinix, y hi sik 
2A ae 
cant s Ene ean a ons: 


a 
Persent of Semi-periods 


Fig. 389. The Two Ways of Straightening Out Semi-cycles of a Sine 


Curve. 
se 


PE ECi abi PROJECTED SCALES 487 


ae 

Tt is a rule of general application that any curve can be 
easily made straight if it does not undulate, that is, has 
neither peaks nor valleys. The process of straightening re- 
quires no more than the projection of the y-axis scale, that is, 
the scale of the independent variable, in such a way as to 
make its calibrations record the values of the ordinates at 
equal intervals along the x-axis scale of the arithmetically 
projected chart. A more detailed specially projected scale is 
given by the following steps: 1, Divide either axis of the arith- 
metically projected chart into uniform parts or intervals; 2, 
From the points so obtained erect ordinates or abscissae to the 
curve, and from their intersections with the curve project ab- 
scissae or ordinates to the other axis, and read the values 
thereon; 3, Lay off these values along the first axis but calibrate 
them with the original values. The object of this is always 
the same, namely, to introduce in the altered scale of the 
variable those inequalities and irregularities which will exactly 
counterbalance the irregularities of the curve and smooth it 
out into a straight line. 

It will quickly occur to the student that such specially pro- 
jected scales can even be made for some undulate or periodic 
curves, to reduce them to regularity and uniformity of undula- 
tions. By undulate curves we mean curves with peaks or val- 
leys or both. Thus a sine curve can be laid out on such spe- 
cially projected scales so that it forms a succession either of 
semi-circles or of angles and straight lines. It is in such work, 
however, most obvious that the peaks should be correctly 
positioned at the maximum ordinates or abscissae, the valleys 
at the minimum ordinates or abscissae and where cycles are 
different in amplitude or phase, that they be converted to com- 
mon levels or intervals. The intricacy of this work is consid- 
erable, its usefulness highly specialized and the economist, 
sociologist, or business statistician will have little occasion 
for it. 

The subject of special projections and their use for the 
analysis of curves is still in an elementary stage and little can 
be dogmatically stated about it. Collections of the typical 


Nore To Fie. 389 


The first requires the conversion of all amplitudes into percentages of the nodes, 
and permits the use of variable phases and periods. The second requires the con- 
version of all cycles to common units of & or percentages thereof, and permits any 
amplitude readings. 


. 488 CHARTS AND GRAPHS 


hyperbolic, parabolic, and other curves have been published, 
intended as guides to the engineer to assist him in the recog- 
nition of the nature of given curves. Such collections famil- 
iarize the student with the curves of his equations, but are re- 
stricted in their usefulness in the reverse process of equating 


COLD STORAGE HOLDINGS OF EGGS 
Average Monthly Stocks of "Case Eggs" in Warehouses 
United States 
Average of five years, 1916-1920 
(Source:- Survey of Current Business) 
(Relative figures, 100 = average month) 


z Cond fo} ow nn ~ 
Brooke $ g & $2 8 3 


200 


Percent of Averege Vonth 
5 
° 


60) 


Dec 
Jan 
Feb 
Mar 


Months 


Fig. 390. Showing How Closely the Cycles of One Set of Periodic 
Economic Data Approach a Sine Curve Wave. 


curves. The trouble arises in the fact that slight changes 
either in the scale of the chart or the constants in the equation 
produce great changes in the appearance of the curve. It is 
therefore impossible to prepare complete catalogs of curves 
classified by their shapes. 

It is probable that in the development of special scale pro- 
jections which straighten or regularize the curves of certain 
equations, the greatest advance in the science of curve equat- 


OP ECEALEYs PROJECTED SCALES 489 


ing will be made. For very often these special scales are free 
of the limitations of the fitted curve, neither scale alterations 
nor changes of constants affecting the regularity of the curve 
upon the chart with the correct special scale. This is a field 
of graphics as yet little explored, its possibilities are just 
opening up to us, and only time and continued usage will de- 
termine what forms are valuable and the limits of their value. 
It is clear, however, that the probabilities projection is not 
the last invention of its kind nor yet is the square or powers 
projection, 


CHAPTER XLIV 
FORMULAE FOR CURVES 


While it is a great step forward in the analysis of data to 
have plotted the curve and be able to visualize the behavior 
of the phenomenon, yet the mathematician often seeks to 
take a further step, and formulate from the curve a law for 
the data, by which its behavior will be precisely described in a 
single mathematical sentence. This mathematical sentence is 
known as an “‘equation.”” And it is the object of the equation 
to give us a general description of the data under all circum- 
stances and times for which the equation is prepared. The 
process of describing the behavior of a curve with this math- 
ematical precision, is called writing an equation to the curve. 

For a great many curves the description is extremely easy 
to formulate. If you see a straight line on amount-of-change 
paper, in which the two scales are alike and the straight line 
slopes at an angle of 45° from the origin of the chart, you will 
say at once that the y-values which the line passes through 
are equal to the x-values, or that the y-values are increasing 
equally with the x- -values; and you would express this de- 
scription mathematically in the sentence (or equation) y =x, 
that is, that the value of y for any x is the same as the « itself 
and whatever the x-value is, that also will the y-value be. |Let 
us vary the case a bit. If the y-scale be twice as great as the 
x-scale, that is, each unit on the y-scale equal to two units on 
the x-scale (the line still sloping at 45 degrees), it would not 
take you long to determine that the formula or equation for 
the straight-line curve is y =2 x. Again, let us suppose that 
instead of passing through the y-axis at the origin, the curve 
passes through it at the value of 3. Now this adds 3 to each 
value of y throughout the length of the curve and so you will 
quickly write the formula as y =2 x+4+3. 

Or let us take a descending curve which on a chart with 
equal scales on each axis describes a 45° downward slope from, 


490 


FORMULAE FOR CURVES 491 


2 

Cou Se 

ete Leb [es (aN 

CP RC ar ae er "012834567 6 9 wWili2 
Fig. 391. The Linear Equation, y=ax-+c. 
EXAMPLES: 

y=x y= 2x 
y=2x+3 y=10—x 


The value of c can be read at the intersection of the curve with the y-axis; in 
other words c¢ is the ordinate of the curve when x=o. The sign (+or—) of a is 
shown by the upward or downward direction of the curve; its value can be found 
from the phrase, a=y—c, when x= 1. 

(Note.—The small letters immediately above each diagram in this chapter indi- 
cate the functions plotted to form the curves.) 


let us say, the point of 10 on the y-axis. A little study will 
show you that as the curve descends, the values of y diminish 
from their original value by the amount of the corresponding 
x-values. This condition can be expressed mathematically by 
the sentence, y =10—.. All of these cases are simple and we can 
observe that in all of them the curve forms a straight line. So 
we may hazard a guess that whenever we meet a straight line 


492. CHARTS AND GRAPHS 


upon amount-of-change paper, we can write an equation to it 
which will be a simple equation or an equation of the first 
order, that is, both the unknowns, y and x, will be found in 
their first powers. The general formula for the straight line on 
plain co-ordinates is sometimes written as y =ax-+c, in which 
both a and ¢ are constants for the particular line. (The values 
of these constants were seen in the examples just given to have 
been, for. “a, by 23i2, ane 15 andifogac. 0... alee) 

Remembering that the logarithmic projection substitutes 
the processes of multiplication and division for the processes 
of addition and subtraction, we may further generalize that a 
straight line upon logarithmic paper will have the general 
formula of logy =) log «+d, or log y =) log x+log a, which is 
the same as saying y=ax’. And a little experimentation will 
show you that every equation of this form (a and b being con- 
stants) will appear as a straight line upon a logarithmic chart. 
Here again we find a simple formulary relation which it is 
convenient to determine. For an equation so simple as y= 
ax +c or y =ax’ is distinctly more convenient to remember and 
apply than the plotted curve itself. Two familiar examples of 
this are to be found in the computing of interest, the first 
being the formula for simple interest plus principle and the 
second for compound interest plus principle. 

The special scales which have been discussed in the pre- 
vious chapter are of course designed expressly to whip into 
straight line formation the recalcitrant and unwilling curve, 
and when they succeed, or even very nearly succeed, greatly 
simplify the writing of equations. But as has been pointed 
out, they can only be used with care, since some curves, or 
short portions of curves, will behave similarly upon several 
projections, and the approach to a straight line upon one scale 
projections does not always indicate that the formula for that 
scale is the best, or even a correct, formula for the curve. This 
danger has already been mentioned and illustrated. 

A little study of the various special scale projections will 
show that they are all outgrowths of the simple linear equation 
of the straight line upon uniform or arithmetically projected 
scales. ‘The object of the special projection in each case is to 
so graduate the values of the scale that they absorb all the 
powers of the variables, leaving to be plotted the remainder of 
the equation, in which the variables occur in the first powers 
only and which therefore form straight line curves on the 


FORMULAE FOR CURVES 493 


chart. Lipka enumerates eleven typical equations whose 
curves straighten out upon the charts with the scales and we 
shall briefly repeat this list, that the student who has found 
a combination of scales which makes his curve straight may 
quickly find the equation best describing his data.1 

For the straight line upon uniform (that is, arithmetically 
projected) scales, we have an equation in the first degree, 
called, from its form, the linear equation. Its type is y=ax-+e, ° 
in which a and c are constants whose values can be easily found, 
c being the intersect point of the curve upon the y-axis and a 
being the tangent of the angle of the curve upon the x-axis, 
easily computed from any observation after c is known. For 
all curves which pass through the origin, the equation is re- 
duced to y=ax, since c has disappeared. 

For the straight line upon log paper, both scales being 
logarithmically projected, we have, as we have seen, the equa- 
tion log y=b log «+d, which is but another way of saying 
yy =ax’ in which a, b, and d are constants, d being the logar- 
ithm of a, and both d and D are as easily found as c and a above. 
Drawn upon arithmetical paper, the curve is, of course, not a 
straight line, but becomes a simple parabola or hyperbola. 
It is a hyperbola, that is, in this case, a falling curve, if b is 
negative; if b is positive, the curve is a parabola, that is, in 
this case, a rising curve, and approaches the vertical as it in- 
creases if b is greater than unity and approaches the horizontal 
if b is less than unity. In the one case where 3 is unity, the 
curve straightens out (on arithmetical paper) since here b can 
be omitted from the equation and the latter becomes y =ax. 
Thus we see that the equation y =ax will be a straight line upon 
either the arithmetical or the logarithmic projections. This 
is the same as saying that if a straight line curve on arith- 
metical paper pass through the origin, it will also be a straight 
line upon logarithmic paper. A 

We have in the previous chapter mentioned the shifting of 
the zero point upon a logarithmic projection, that is, the re- 
calibration of the scale after it has been graduated (plotted.) 


1 The remainder of this chapter is largely and very inadequately drawn from Pro- 
fessor Lipka’s excellent book, Graphical and Mechanical Computation, John Wiley & 
Sons, 1918. This has been done not to substitute the present volume in any way 
for that treatise; it has rather been the writer’s purpose to draw attention to the 
extraordinary possibilities opened up by Lipka’s work, and to direct readers to it. 
The volume is indispensable to the student, and to the technician it will almost certainly 
open up a new world of research, arming him with invaluable implements. 


KY 


biel ee ab 
wy 


eed 
sie 


ZS 
al 


Cae mee Ss 


sien 


SE Fd 
ee an iia 


Ll ie 
bel eal 


am 
ES = 


Va 
: 
= 


ee ee 


log xy log y 


NI | 
ina 


a 


10 


or ee Sek b 1 log Xe 


aah 


See footnote on opposite page. 


Fig. 392. “The Curve ¥ y 


FORMULAE FOR CURVES 495 


The scale then represents, of course, log (y —c) where ¢ is the 
constant which has been added to the plotted values to give 
the calibrated scale-figures. When such shifting of the scale 
has straightened out a curve upon logarithmic paper, the curve 
has, of course, the equation log (y —c) =b log «+d (instead of 
the immediately foregoing log y =) log x+-d). From this new 
equation we derive y —c =ax’ and so y=ax’+c. Thus we see 
that the shifting of zero-points on the log scale is but an ad- 
justment which makes c disappear and gives the straight line 
on log paper. On log paper without shifted zeros, the curve is 
parabolic, concave to the x-axis if c is positive, convex if it is 
negative.2_ When ¢ is zero it disappears from the equation and 
the latter becomes y=ax*, an equation already described, 
having the straight line form on unshifted scales of log paper. 
And when bd becomes unity it disappears from the equation 
leaving an equation of the first type, forming a straight line 
on arithmetical paper. 

The three types of equations are closely related, all having 
the general form y=ax’+c, in which Dd is unity for the first 
type and any number for the others and c is zero for the second 
type and any number for the others. The third is distinct 
from the first and second in that it contains not one or two, 
but three constants, and hence requires calculation (when we 
are seeking to find the proper shifted scale) for the third con- 


2 The value of c can he computed by taking, any three items in the data (or points 
along the curve plotted experimentally or. uniform paper) such that their x-values 
form a geometric series, 1.¢., %1:%20:%2-%3. Then 


x= Vx1%8 
and axeb=~/ ax? ax? 
Hence ye—e= V (yi—¢) (ys—e) 
ViV3— Vo 
and gre yitys —2y2 


So we must observe the ordinates, 41, 2, yg at these points and substitute them in 
the last equation to get the value of c. 


SCALE PROJECTIONS OF FIG. 392 

Arithmetical 

Logarithmic 
The value of a is shown by the ordinate of the curve when x=1 (i.e., log x=0). 
The curve is hyperbolic when J is negative and parabolic when it is positive. 
(When 5=0, the curve is a straight line parallel to the x-axis. If b=1, the curve 
is also straight upon the arithmetical projection, its equation, y=ax being linear 
(see the curve y=x). The value of } can be found from the phrase, b=log y— 
log a, when x=10 (i.e., log x=1), 


CHARTS AND GRAPHS 


=A 


Fig. 393. The Curve of y=a’+c or, log (y—c) =log a+6 log x. 


EXAMPLES: 
SCALE PROJECTIONS: 
Arithmetical Logarithmic 
Logarithmic with shifted zeros 
The scales of log (y-c) in the two lower diagrams are logarithmic projections with 
the scale-figures altered by the value of c; when c is negative the scale of log 
(y-c) will include the value of zero. The value of c must be known before the 
scale can be so altered. Four curves such that they straighten out on these 
shifted scales, are shown, two curves for each scale, one ascending and the other 
descending. ‘These four curves are also shown on other projections, showing 
their various shapes. The value of a can be found by the phrase, a=y—c when 
*=1 (i.e., log x=0). The value of b can be found from the phrase, b=log (y—c) 
—log a when x=10 (i.e., log x=1). 


FORMULAE FOR CURVES 497 


stant, c. For the third constant is not readily capable of 
graphic solution, save on arithmetic paper; it must always be 
known or mathematically calculated before a proper altered 
scale can be found. Of course when we have by experiment 
found a satisfactory scale it amounts to a trial and error 
method of solution. 


We also noted in the previous chapter the projection of 
powers of variables along the scale, the use of the square-root 
projection being illustrated. In these scales the values of x 
have been entered as calibrations or scale-figures at points 
which were plotted or graduated for the values of x*, “k”’ being 
the known exponent of the power. A curve therefore which 
straightens out upon this when the y-axis is uniform (arith- 
metical), has the formula, y =ax*+c, in which the constants, 
a and c, are found as before. This equation is of the same 
general type as the foregoing, y =ax'-+<c, its only difference 
being that d is a known, not an unknown constant and is 
called k therefore. Here D or & plays the role of the third 
constant, being known. And when D is a small positive in- 
tegral, such as 2, this scale projection affords a simple means 
of straightening the curve, but unlike c, b cannot be easily 
calculated when it is unknown. The method therefor is lim- 
ited to the use of curves in which b is known, and is valuable 
in such cases when Dp is a small integral. It is much easier, for 
example, to prepare a squares projection than to shift the zero- 
point on a log scale. 


The squares projection is an example of the powers projec- 
tion in which the exponent of the power is a positive integer. 
If on the other hand, the exponent be fractional, we have an 
inverse power or root. An example of this would be a square- 
root projection. Or the exponent may be negative. The 
simplest instance of this is the reciprocal projection, for the 
reciprocal of a number is its —1 power. When a curve straight- 
ens out on a chart one axis of which is reciprocally projected 


(the other arithmetically) its formula is obviously y ete 


a modification of the general type formula y =ax*+c (in which 
k= —1) or y=ax’+c (in which b= -1). Every powers pro- 
jection, therefore, when used along one axis only, always 
straightens out a curve whose formula involves that partic- 
ular power of the one variable. We may treat all the possible 


vw 


xy 
————— 
| 
+ pe. ae 
t ez 


aekaADAADwWO 


Fig. 394. The Curves of Known Powers, y=ax'te and, 
See footnote on opposite Page. 


y* =ax-+c, 


FORMULAE FOR CURVES 499 


powers projections as but one class, with formulas of the 
type y=ax*+c. These formulas contain three constants, only 
two of which, however, are unknown. The third constant is 
known, and is the exponent of the variable. And surely it is 
clear that whenever the power of a variable is known, that 
power may be laid off upon the scale for that variable so that 
the plotting of only the first power of the variable (that is, 
the variable itself) thereon, will make the curve a simple linear 
one. The power remains in the scale and hence in the formula, 
but has vanished from the curve. ; 
In all powers projections so far considered, we have used 
the special projection upon one scale only, the other being 
uniform. The equation being y =ax*-+c, it is clear that the 
x-scale has been specially projected, for the given power, k, of 
the variable, x. Care must be taken to keep this arrangement, 
for a reverse arrangement will fail to straighten out the curve. 
Only i in the case when c =0 and the equation reduced to y = ax", 
is it immaterial which scale be subjected to the powers BEES 


tion, for here we may write y =ax* or Wate a’x, in which a’ = 


as The line will therefore be straight either upon the 
powers projection of one scale or the corresponding root pro- 
jection of the other. yy =ax* is, however, too easily straight- 
ened out by the log projections, as we have seen, and hence 
the case is of no value. The real use for the powers (and 
roots) projection of the y-scale is in the wholly different equa- 


tions of the form y* =ax+c (including yea are). 
Closely related to the projection of reciprocals, is the pro- 


5 O e a e 

jection of products. Thus the equation y=-—-+¢ may be writ- 
x 

ten xy =a-+cx. In this form we see that the equation will not 


; Me age 1 
only straighten out upon semi-reciprocal scales, —, y, but also 
x 


SCALE PROJECTIONS FOR FIG. 394: 


Square of X. Square of Y. 
Arithmetical 
Square root of X. Square root of Y. 
Reciprocal of X. Reciprocal of Y. 


Six typical curves are shown by full lines, one on each of the specially projected 
scales and all upon the arithmetical projection. They straighten out only upon 
the scales on which they are plotted. If, however, c=0, that is, there is no added 
constant and the equation is reduced to y=ax*, then the curves are straight 


upon either of two different projections, thus on x’, y or % “W/y;on x, y? or vx, 
y,; and on 1/x, y or x, 1/y. Such curves are Sowa by broken lines. 


AS28358 5233 


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QO OFS Baa wi aA ral Moe 6 oF © 2g 


x 


CHARTS AND GRAPHS 


500 


o> ot 

Jig ive 6 
g ee"s 8 
4 os é 
9 usrz 9 
@ sez 9 
; oz» 
¢ ist ¢ 
Z set Zz 
T or t <a 
= ust ° : w 
T Geo ater os 
ze o 2 
ba get- gp 
iad ae oa 
cm ‘Ie ge 
9- es't- 9- 
zy, eoet= s= 
8- ea ee 
re ss'z- 6- 
ot £9°Z- Ot- 
$ © 

Ki om 
PP . s hace 1-1 3° a a a rl | ra < 
9° - 4+ +- 9 
L tH ea : 
8° 7 8 
6 i 
a oe os lez im ot = + ay t 
21 a | it LT art Ao “ 
$1 ae 11 9°t bay + 
2 vt 2 5° : H z £0 x id 
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© : oo 9 Tt ; e 
ot- Chey cy | 4 ot 
eS NESSES Se oe 
a ~~) eas re mt a at ik z- 
#t | es 
2°I- Bt=. 
Se ee ee sat ee 5th fe Sea | L . 
« 


x 
a+ex 


+e and y 


a 


x 


See footnote on opposite Page. 


Fig. 395. The Hyperbolic Curves, y 


FORMULAE FOR CURVES sol 


upon the semi-product scales, x, xy. Product and quotient 
scales are, however, a little hazardous, in that the introduction 
of one variable into both scales may often force a curve to ap- 
proach nearer to, though not entirely to, a straight line, with- 
out the least real significance. Moreover, they require some 
computing, as the series of y original data must be replaced 
by the series xy, in which each value of x is multiplied into its 
corresponding value of y. To be sure, no special scale need be 
projected, the products or quotients being put upon a uniform 
(arithmetically projected) scale. 


| The most interesting use of the reciprocal projection is for 
the equation in which both variables are in reciprocal form, 
namely y-1=ax~-1+c. This is the equation of the ordinary 
hyperbola. On uniform scales, it is asymptote to the co- 


EXAMPLES AND SCALE PROJECTIONS FOR FIG. 395: 


Single Reciprocal Product, xy 
3 
ree a xy =3--4x 
Double Reciprocal Arithmetical 
eee 
—-= —+ ee Both 
y= Se 83 
Quotient, x/y Quotient, y/x 
x zy C4 2y 
ra SOE haa 


8 
Two typical equations of hyperbolic curves are shown here. The one shown by 
the broken line, involves the reciprocal of one variable only and straightens 
upon a single reciprocal projection; from it a series of products of the two vari- 
ables can be computed which straightens upon arithmetical paper on the plotting 
of x and xy. Note that its asymptotes are 0 and c; and that the value of ¢ can be 
found by inspection at the intersection of the curve with the y-axis, calibrated 
as infinity, on the reciprocal projection. The value of a can be found from the 
phrase, a=y—c when x=1. 
The other equation, shown by the full line, involves reciprocals of both variables 
and straightens out upon the double reciprocal projection; from it two quotient 
series can be computed which yield straight lines. Note that its asymptotes are 
y=1/e and «=—a/c and can be read at the intersections of the curve with the 
axes, calibrated infinity, upon the reciprocal paper; from which the values of ¢ 
and a can be easily found. 
(Note.—The small letters over each diagram in this chapter show the functions 
(of the variables) plotted. Large letters, X and Y, indicate the scale-figures or 
calibrations, and are omitted if these are graduated for the same functions; the 
curve can then be plotted directly from the scales without finding the functions. 
The presence of the large letters indicates that the curve cannot be directly 
plotted from the scales, that the indicated functions must first be found and these 
(instead of the variables) must be plotted from the scale-figures. ) 


502 CHARTS AND GRAPHS 


ET EN LTT TTT 
TTS TT 
He | 
PT 


167 


wa See SB SUE GSS SE A ey Se SY eae ge Sane 


z 
~ 
8 
6 
9 
+5 10 
lL 
3 18 
o 


“4 4 


Fig. 396. The Hyperbola with Three Constants, y= Geert 


SCALE PROJECTIONS: 
Arithmetical. Double reciprocal, one shifted. 
Quotients, singly shifted. Quotient, doubly shifted. 
A single typical equation is shown in hyperbolic form on arithmetical scales and 
straightened upon reciprocal scales, one of which has a shifted zero. The re- 
ciprocal scale with the shifted zero can be prepared only when the added con- 
stant, d, is known, since the zero is shifted by its amount. Also when dis known 


x y—d 


the quotient series, ay or ——, can be computed from the data and will 


y—d 


: ; ;: x . 
yield straight lines upon the plot of either, x, Sy OF , ». When d is un- 


—d 


FORMULAE FOR CURVES 503 


; a 1 : 1 
ordinates x=. and y=—. From its form, -=%+c, we can see 
c c me 
that it will straighten out upon paper in which both scales 


are reciprocally Bip yecrerae: If for any reason we desire a 


chart giving detail to different parts of the curve, we can re- 
: : x 
write the equation as pe otee and then we see that by the 


use of the scales, x, r in which both are uniform and one, the 


y-scale, is used for the plotting of the quotients of the values 
of x by their corresponding y-values, the curve can again be 
straightened out. Or we can cast the same equation into the 


1 


: a ay 

ea sy OF 1 a be OR and so see that the curve will 
. » 

straighten out upon the scales, y, ms Here are three arrange- 


ments by which the ordinary hyperbola can be straightened 
out. Take your choice. The constants ‘‘a’’ and “c’” can be 
found by inspection from the plotted curve on the double re- 
ciprocal paper. 

x 
at cx’ 
will occur to the student that it may occur in modified form with 


If we write this equation in its usual form, y= 


a third added constant, thus ye es +d. The curve of 


this modified equation will still be hyperbolic upon uniform 
paper, but will no longer pass through the origin of the chart. 
Just as in the first and third equations discussed, so here the 
added constant may be determined by the ordinate of the 
curve at the zero-point on the x-axis, that 1s, the point where | 


known, the zero must be shifted along both axes to some known point in the 
curve, and if we call the co-ordinates of this known point (it may be any we wish 


to select) xo and yo, then we can compute the quotient series which 


—Yo 


xn—x * . 
yields a straight line y upon the plot of x, ae The value of d is obviously 


shown by the intersection of the curve with the origin, that is, d=y when «=0. 
Note that the asymptotes, y=1/c+d, and x= —a/c, are shown upon the recip- 
rocal projection by the intersections of the curve with the axes, calibrated infinity; 
from this the values of c and a are easily found. 


504. CHARTS AND GRAPHS 


the curve intersects the y-axis upon arithmetically projected 
scales. This cannot be read upon the reciprocal scales as the 
latter never reach zero, since zero would have to be plotted 
at its reciprocal, infinity, a manifest impossibility. We must 
therefore first plot the curve on uniform scales to determine, if 
we can, the value of d by inspection. Then if we write the 


Aue x a 
equation in the form y At tae and then vaaE Go +c, we 


shall see that the curve will straighten out upon the reciprocal 
i sh . 
scales ead In short we have now come to the use of 


shifted or false zeros upon the reciprocal scale. The curve can 


, and 


also be straightened out by the quotient scales, x, 


y, > ct . In every case there is a great deal of computing to 
os 


be done. 


When the value of d is not easily found, it may be advis- 
able to use a method of differences, which has not so far been 
mentioned. Virtually this amounts to shifting the zero-point 
(or origin of measurements of the co-ordinates) to any con- 
venient point we wish along the curve. To do this we first 
select a point upon the curve or an item in the co-ordinates 
which we may indicate by x) and yo. Then we compute the 
difference between these co-ordinates and all other co-ordi- 
nates, x and y, in the data. Finally we divide the differences 


5 , ts R 

to get the series of quotients “and plot these as co-ordi- 
Vs 

nates over the aPaceras of x. The reason for this is that the 


equation, y=—* at, +d, can be reduced, by subtracting the 
x 


selected point, y.= 


“d,/ 6 tiie form =a+cx tex 
aus y —o 


nada The added constant, d, which has caused all the 
a 


trouble, has been, as you see, eliminated, and the constants, 
a, c, and x», are left. Write the second half of the equation 


as (a+cx») ate (a+cx,) x, and you will see that the curve will 
“i ane 
= Vib 


Of course this is not 


straighten out on the plot of x, i 


FORMULAE FOR CURVES 505 


a plot of the original series, it is only a plot of the cotangents? 
of the points upon the curve from the selected point in the 
curve, but if the derived curve be straight it is proof that the 


a+cx ams 
In all reciprocal scales the constants a and ¢ are easily found 
by inspection. The axes of the scales are always calibrated 
as infinity along each scale and cross the curve at its asymp- 


totes. The asymptotes have the value x=“ and (y -d) = iy 
c G 


original curve has the formula y = 


So we need only substitute the observed values for « and y to 
obtain the constants and complete the formula. 

Similar methods can be used to straighten out the ordinary 
parabola, the equation for which is y=a+bx+cx®. (The 
reader will note that to arrange the variables in ascending 
order, we have altered the symbols for the constants, and 
that this equation is really a modification of the simple linear 
one, y=ax-+c.) A little study will show that the value of 
the constant a is the value of the curve at its intersection, ex- 
perimentally drawn, with the y-axis. Since the equation can 


3 A) re 4 F 
be written > —*=b-+cx, it is obvious that if we know a, we 
a 
: : =a 
can compute the series of quotients Y=“ and that the curve 
a 
: ; =—d. 
of these will straighten out upon the scales, x, 2 DoT firwe do 
a 


not know the value of a, we can use the method of shifting 
the origin to a point upon the actual curve! and will get a 
straight line by plotting x, Y=" (nor ~=** because recip- 
x =X y Yo 
rocals are not involved in the equation). If the values of x 
in our data form an arithmetical series, we can take the suc- 
cessive differences of the y-values, that is, A y, and will find 
that the plot of x, A y,is a straight line.5 The polynomial in- 
volving higher powers sych as y=a+bx +ex?+dx3+ .. . must 
be successively differentiated or the method of determinants 


3 Cotangents only because they are reciprocals. 
4 By substituting yo=at+bxotcxe we get y—yo=b(x—x0) +c(x?—270) or 


ee  hthcry tex, in which the phrase (¥-+cxo) is a constant. 
xX—Xy 
5 The formula for A y is to be found in Lipka, “Graphical and Mechanical Com- 
putation,’ p. 146. 


CHARTS AND GRAPHS 


506 


3 2-4 
X=%, Ve oe pie 240 


OY 


Y-¢ 


Y-+ Yo 


a+bx+ex?. 


The Parabola, y 
See footnote on opposite page. 


397 


Fig. 


FORMULAE FOR CURVES 507 


“used, but the riathemetice in such work is outside the scope of 
this aliens Indeed, the phrase cx? may be added to any of 


the foregoing equations and will produce the same difficult 
results. 


The close observer may note that so far no mention of the 
semi-logarithmic chart has been made. The curve which 
straightens out upon it is known as the simple exponential or 
logarithmic curve. The formula of such’a curve is of a wholly 
different nature from any so far considered, for in the latter 
all exponents have been constants. In the equations of expo- 
nential curves, we meet with variable exponents. The simple 
exponential curve has the formula, y=ab?. The curve upon 
arithmetically projected scales is convex to the x-axis, rising 
rapidly as x increases positively, and having a horizontal 
asymptote to the x-axis as x increases negatively and inter- 
secting the y-axis, at the value of a, since “‘b”’ becomes unity 
at the y-axis (x being zero there and the zero power of any 
number being one). Now if y=ab*, then of course log y= 
log a+x log b=a'+b’x (in which a’ and b’ are constants, the 
logarithms of a and b). The second part of the equation be- 
comes familiar enough in this last form, being the very first 
type considered. So if we plot the curve upon semi-logarith- 
mic paper, x, log y, it will straighten out. And conversely all 
straight lines upon semi-log paper have the equation y =ab’. 
And this formula applies to all historical data which have 
straight-line curves upon semi-log paper. It is the formula 
of the law of organic growth. 


Several variations of the exponential curve equation are 
obviously possible. If a constant be added we have y =ab*+<. 
This we turn into y —c =ab* and from it we derive log (y —c) = 
log a+x log b or log (y —c) =a’+b’x. Here we see a need for 
the shifted zero upon the logarithmically projected scale, an 
the curve straightens out upon the scales x, log (y —c). If 
c is unknown it can be computed from the experimental curve 


"SCALE PROJECTIONS FOR FIG. 397: 

Quotients (shifted) and differences. 

Arithmetical. 
The ordinary parabola (shown by a full line on the lower diagram) is the sum of 
three different curves (shown by broken lines) and cannot be straightened out 
upon any useful projection. It can be made to yield, however, various straight 
lines for series which have been computed from its known data and these afford 
a test of its equation. 


508 CHARTS AND GRAPHS 


x 


, log (y-6) 
Pere 


oon or ow = MPH nO eaeweweonr @aeo 
Re TAT wee te Ceinta bo) to sate See Sor LOL SER SU ae tee h a 


x, log Y= log Ve 


a2 

21 

20 + ei | zi ~~ 
19 

zs FA ia 


Fig. 398. Exponential or Logarithmic Curves. 


EXAMPLES: 

y=ab*, or log y=log a+x log b 

y=ab*-+c, or log (y—c) =log a+~ log b 

y =ab*c*, or (1/x) log (y/a) =log a+x log b 
The simple exponential equation y=ab* straightens out upon semi-log paper. 
When a constant c is added, the zero must be shifted by the amount of the 
constant on the log axis; one example of this (shown by the broken line), is plotted 
upon three diagrams to show its behavior, another which on arithmetical paper 
parallels the simple equation shown, is plotted on two (shown by the dot-and- 
dash line).. These curves are hyperbolic. When a higher power of the variable 
is added, the curve becomes parabolic and cannot be straightened. A quotient 


log y—log a ; 
oo 
which will yield a straight line; this line (shown dotted), is here plotted upon 
Scan 


series with shifted zeros can, however, be computed, namely, 


semi-log paper as aes simply for the sake of variety. 
a 


FORMULAE FOR CURVES 509 


upon uniform paper.6 Another exponential curve is that which 
has the formula y=ab*c". If it be written as log y =a! +b'x+ 
c’x? (in which the primes of the constants again represent their 
logs) we see that it is very similar to the ordinary parabola, 
and indeed, precisely the same methods must be used to 
straighten it. If a is known, we can plot it upon the uniform 


log y -log a 
x 


scales x, ( ). If ais unknown we can use either 


of the other two methods, and derive series whose curves 


, lo —log vy? 
straighten out upon the scales x, | and '2, 


—Xo 


A logy. Other exponential curves have still other formulae, 
which are often but modifications of any of the foregoing 
through addition of other variable powers, such as d* in the 
equation y=a+bx-+cd*. These more complicated equations 
must be subjected to even more devious calculations before’ 
derived series can be found which straighten out and prove 
the equation. 


The reader should not consider from this brief summary of 
the scale projections which straighten out non-periodic curves, 
that all or even nearly all curves. can be straightened out by 
them. And the non-mathematical reader will doubtless have 
a wholesome respect for the processes of curve equating even 
by the above methods. He will probably find little difficulty 
with the simple linear, the simple parabolic and hyperbolic, 
and the simple exponential curves, requiring as these do only 
the arithmetical, logarithmic and semi-log charts. But some 
curves are immensely difficult to express in equation form, and 
must often be broken into parts with separate equations for 
each part. It is true that these parts can be collected with 
proper mathematical symbols of limits, into a single equation 
and in this sense it is true that an equation can be written to 
any curve in the world. 

But the iong and complicated equation has little value. 
The equation for very irregular curves—sucn as the profile of 
a man’s face—may take up more space than the curve itself. 
The disadvantages of complicated formulae are many. For 
one thing, a very complicated formula is difficult to understand 


ee : ; 
6 The same method for finding c can be used as before, for the equation y=ax" c. 


510 CHARTS AND GRAPHS 


even when it has been stated—the average person still has to 
plot its curve to understand its meaning. For another thing, 
very complicated formulae suffer from the danger of being 
made unnecessarily detailed or intricate by chance variations 
in the observations which form the data. 

This last consideration, the danger of chance variations in 
the observed data, leads us to the thought that the “true 
curve” for the data, if all errors were absent, might be a very 
simple curve, easily expressed by an equation, while the curve 
of the actually observed data remains irregular and compli- 
cated. We therefore oftentimes have to be satisfied by simple 
curves which closely approximate the actual curves, when 
such simple curves can be found. And the problem then 
becomes one of “‘fitting curves”? with the best possible (that 
is, the closest fitting) straight lines, in the attempt to find 
simple and approximate descriptions and equations. 

The reader will have seen by this time that much of the 
care expended on proper curve plotting has for its purpose the 
clear visualizing of the phenomena, but that still other care is 
expended in the attempt to capture the curve in a symmetrical 
or regular formation. And he will now see that one of the chief 
purposes of symmetry and regularity is to enable us to formu- 
late laws governing the behavior of the phenomena repre- 
sented by our data and curve. In the discussion of fitted 
straight lines, which is so far as it seems desirable to enter 
the subject in this book, he will be reminded of the “‘trend”’ 
and “secular change’ discussed previously in historical curves; 
in fact for historical series the secular trend is often con- 
sidered to be a fitted straight line. And he will now also see 
that these secular trends can be expressed mathematically in 
equations. He will also see that the operations of interpolation 
and extrapolation can be even more precisely performed when 
the equations are used than with charts only. He will see, in 
short, that the possibilities of mathematical description or 
summarization of curves opens up to him a valuable adjunct 
to the use of the curves themselves. 


“tes 


sida ‘ _ » itone 
ya oe 


7 rane 


sia ian Fae» oe 


eee Pe gna 


ee 4 


ti ude, ] See 


CHAPTER XLV 
CURVES FOR FORMULAE 


Having seen something of the way in which formulae or 
equations can be written to curves, we can reverse the process 
and prepare curves to illustrate formulae. In this way, we no 
longer seek the mathematical statements describing a curve, 
but we seek the curves illustrating a mathematical statement. 
The advantage of writing an equation to a curve lay in the 
fact that, from the equation alone, we could, by mathematical 
operations, find the values represented by each or all of the 
plotted points along the curve; the advantage of drawing 
the curve to illustrate an equation lies in the fact that without 
bothering about the mathematical processes, we can read the 
values represented by the equation directly at a glance from 
the chart. In short, the chart may be made a substitute for the 
processes of calculation and computation, and the chart then 
becomes a calculating machine. 

If, as in the previous chapter, we have a curve for which 
the mathematical equation is Y =2’ X +3, and we wish to find 
the value of Y, when X, let us say, is 5,,we do not have to 
solve the equation by mathematical processes, multiplying 
5 by 2 and adding 3, but from a glance at the chart we can see 
the Y-value of that point on the curve whose X-value is 5. 
We follow the ordinate from 5 on the x-scale up to the curve 
and from the intersect point (where the curve passes through 
or intersects the ordinate) we follow the abscissa or horizontal 
co the y-scale and read 13, the answer. In this case, it is true 
that the mathematical operation of solving the equation seems 
simpler than the graphic one for the reason that we have 
selected for illustration of the principle a stmple*mathematical 
equation. But you will fnd many complicated formulae and 
equations in which the mathematical operations are far more 
tedious and lengthy than the graphic process. In such cases 
it will be useful for you to be able to construct calculating 


St 


| $12 CHARTS AND GRAPHS 


curves and charts by which mathematical equations of the 
given type can be readily solved. 

The purchasing agent, perhaps, buys in foreign markets 
and must multiply his quotations by the prevailing rate of 
foreign exchange and add perhaps certain local charges in 
this country, before he can compare the values of different 
offers. To interrupt telephone conversations with these 
mathematical operations would perhaps be difficult, but he 
could be provided with a special chart on which he would see 
at a glance the real value of offers without interrupting his 
telephone conversation to the parties concerned, 


Y=2xqs 


1 PPS STS Beil 


1oO ~ Ne #& B® YX OD O 


x 
Fig. 399. 


That a single straight line curve upon an arithmetically 
projected chart-field will illustrate a simple mathematical 
equation involving only two variables in the first degree, we 
already know, for any straight line upon arithmetically pro- 
jected chart-fields has an equation of the general form Y= 
aX +c (in the right side of which a and ¢ are given constants 
' and X alone is variable). We can, however, by a series of 
such straight lines show the equation for two independent 
variables. Let us assume for example that c is a variable and 
call it Z and that the constant a is 2. In other words let us 
prepare a calculating chart for the equation Y=2 Y+Z. As 
we have seen in the last chapter the figure 2 determines the 


CURVES FOR FORMULAE 518 


slope of the straight line curve one if the x-scale is only half 
as great as the y-scale then the slope of the straight line would 
be rigidly 45° to the x-axis of the chart. The added element 


22 Y- 2x 


iz 
a Taian 
MLAS 


x 
Fig. 400. 


Z merely determines the height or position of the straight line 
curve upon the chart; the straight line curve passes through 
the origin of the chart when Z is O and in general intersects 
the y-axis at the value of Z because at the y-axis the value of 
X is O and the equation is Y=Z. Now because Z itself is a 
variable we cannot show the equation by a single straight 
line but must use a series of straight lines, each for different ° 
values of Z and must therefore mark off a scale of Z upon the 
straight line curves themselves. The result is a chart with a 
series of parallel straight line curves which are diagonal upon 
the chart and enable us at once to find the values of Y when 
Y=2 X+Z. To read a certain value, as, for example, when X 
is 5 and Z is 3, we need merely read up the ordinate from the 
point 5, on the x-scale, to the diagonal line or curve marked 3 
on the z-scale, and from the intersect of this particular curve 
with the ordinate, read horizontally across the abscissa to the 
point on the y-axis where we find 13, the answer. 


514 CHARTS AND GRAPHS 


To use this chart for subtraction is very easy, for we 
merely reverse the process and the dependence of the vari- 
ables, saying that if Y=2 X+Z, then Z=Y -2 X. If Y=10 
and X =2, then we read across the abscissa from the point of 
10 on the y-scale to the ordinate from the point 2 on the 
x-scale, and note the value of the diagonal which passes 
through this point, namely 6 on the zscale. It is of course 
not necessary to use whole numbers either upon the chart or 
in the equation for we can easily interpolate between the 


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Bi 2% &% SG DG % 


ZINA ASIA SSS KN? 


actual ruling on the chart to estimate, very closely, the values 
desired, when they are fractional. These subtractive charts 
can also be made to show the difference, not upon the straight 
line diagonal curves of Z, but upon the y-axis itself, by making 
the curves express the general equation Y = Z —aX and making 
the diagonal curve a descending instead of an ascending one, 
as illustrated in the previous chapter. Still another method 
for obtaining the same result would be to carry the chart de- 
scribed in the last paragraph down into the negative side of 
the x-axis. 


CURVES FOR FORMULAE 515 


Indeed the calculating chart only becomes difficult to 
understand when we begin to talk about it. The simple chart 
1s much more easily made than described. Yet it is necessary, 


2 


AAA ATAU : 
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‘BAZ MT As 
BAZ ACA. 
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Z22Y- 2X 
Fig. 402. 


for charts of the more complicated formulae, that the elements 
which go to make up the simple chart be clearly defined. And 
the first consideration of importance is the distinction between 
physical distances upon the chart and numerical values 
assigned thereto. If we call the two axes of the chart (x), and 
(y) and the diagonal dimension (z), then we have at least 
distinguished three different possible places in which scales 
may be projected physically and given numerical values. If 
we indicate the physical distances along these scales, measured 
from an origin-point, as x, y, and x, respectively, and the 
numerical values finally assigned to these distances (i.e., the 
scale figures) as X, Y, and Z, we have a simple means of 
keeping two more details separate in our minds. 

The importance of distinguishing in this way between 
final calibrated values or scale-figures, X, Y, and Z, and the 


516 CHARTS AND GRAPHS 


actual plotted scale-distances x, y, and xz, cannot be under- 
estimated, for confusion at this point will baffle the student 
for the remainder of his work upon calculating charts. It 
is to be understood that the small letters, x, y, and z, are 
merely essentials in the planning and making of charts; they 
do not appear upon the finished work. It is to be understood 
that the large letters, ¥, Y and Z, are merely symbols for the 
different variables in the equation to be calculated. If these 
variables are indicated by other symbols in the equation, then 
the large letters will not appear on the finished work, but the 
accustomed symbols will be substituted for them. The large 
letters are useful in planning the work as they clearly indicated 
the axis or scale upon which the variables will appear. If no 
better symbols are to be had, then the large letters X,_Y, and 
Z, one or all, may be retained upon the final chart and its 
formula. Indeed, even the small letters, x, y, or z, may be 
finally used for this purpose; they then of course indicate the 
variables and scale-figures and have no application to scale- 
distances. But during the stage of making the chart, we shall 
always use the symbols consistently in the meanings specified. 
Thus if we have the equation “income — operating expense = 
operating profit” or “i —e=p,” we shall substitute Y, let us 
say, for 1; Z for e; and X for p; and write Y -Z=X. Whenthe 
chart is finished, we shall substitute the original symbols 
again, and write 1 —e=p. 

But between the scale distances, x, y, and z, and the final 
calibrations, X, Y, and Z, an elaborate structure of modifica- 
tions and substitutions may be built up. These are necessary 
for very complicated formulae; in simple equations they fall 
together like a house of cards and can be wholly disregarded. 
Thus if our equation be Y=X-+Z, we can obviously lay off 
the distances, x, y, and z, directly from the equation. We may 
even use the same face of the ruler for both y and x, that is, 
along the axes of the chart, plotting the diagonals to conform 
to the equation. But when Y=2X+4Z, the chart becomes 
very tall, and as we have seen, it is just as well to lay off the 
Y and X scales differently. 

Since we have occasionally in this way to use different 
units of measurement in laying off scales, it is well to have 
clearly in mind one common unit of measurement for the entire 
chart. This unit we call the “modulus” of the chart; and it 
does not matter whether the modulus be one inch, one foot, 


CURVES FOR FORMULAE $17 


one centimeter, or any fraction of these, so long as it be the 
same for all parts of the chart the proportions of the various 
parts of the chart are the same. The modulus then is simply 
a general unit of distance which serves in planning the chart 
to equate the scale distances and the’scale values; thus, « =mX, 
Orn=2m.X; 


Now it is a great convenience to plot distances directly from 
the data, that is, the values or scale-figures to be assigned. 
When this can be done we can copy scale-figures directly from 
our ruler as we plot. And here secondary moduli for each scale 
become useful. These’are merely fractions or multiples of the 
chart-modulus, and when they differ from the latter, may be 
indicated by m,, m,, or m, Thus when x =mX, m,=m, but 
when «=2mX, m,=2m. In the charts already considered, we 
have seen the chart of Y=2X+Z made with the horizontal 
units of measurement twice as long as the vertical ones. If 
the vertical units be m, then the horizontal ones are 2m. 


Experience will show that it is best to proportion a chart 
in such a way that all intersections be as sharply drawn as 
possible. The object is to make readings from the chart 
accurate. If two lines are perpendicular, there can be little 
doubt about their intersection point, but when they cross at 
small angles (that is, are nearly parallel) it is not so easy to 
decide the exact point of intersection. Since the x and y co- 
ordinates are perpendicular, obviously the z-diagonals cannot 
cut both co-ordinates more sharply than at 45 degrees. So 
the most desirable form of chart is one in which the z-diagonals 
form about 45° angles with the axes. And it is the primary 
purpose of the scale-moduli (not the chart-modulus) to pro- 
duce this condition. When Y=X-+Z and «x and y have equal 
moduli, the z-diagonals, as we know, have the right slope. 
And so when Y =2X+Z it is easy to see that the modulus 
of the x-scale (letting m,=m) must be m,=2m, before the 
diagonals will have the same slope. Here we may note that 


—=2, the coefficient of X in the equation. And it is a use- 
m 


ful empirical rule that the coefficients of the variables (on the 
axes, that is, X or Y) are the ratios of their scale moduli to 
the chart modulus. 

The scale moduli (as distinct from the chart modulus) 
serve still another purpose, for since they form what we might 


518 CHARTS AND GRAPHS 


call “plotting instructions,” they can be used to indicate the 
side of the engineer’s hexagonal rule! which isto be used. Thus 
if we use a chart modulus of one inch, we can plot m from the 
10-side of the rule, 4m, from the 20-side of the rule, 3m from 
the 30-side, and so on. 

From the outset in chart making for formulae, we must 
keep in mind the desirable limits of the variables to be shown 
by the scale figures. If our chart is to be used in calculating 
a few pounds, it would be foolish to make it include tons as 
well, for then the scale for pounds would be sosmall that it could 
not be accurately read. If the price of paper is quoted in 
cents, why make a chart which shows millions of dollars, and 
on the scale of which cents are so small as to be invisible. 
Obviously the larger our scale becomes the more clearly it 
can be read, and the more accurate will be its calculations. 
Hence we should try to include in the range of the scale only 
its useful parts that we may make them as large as possible. 
This calls for the setting of limits f r the range, an entirely 
arbitrary matter, for which it is only necessary that we know 
the extreme high and low values of the variables which will 
be met with in the use to which the chart will be put. Having 
determined these values of the independent variables, we can 
write them into our formula by a convenient trick, thus 

10 12 Ze 10 12 
ot 5 5 0 +s 5 
Now we know how much space to give to the chart, or how 
large to make the chart-modulus for a chart of a given total 
size. 


We are now in a position to consider the havoc wrought by 
constants in a given formula for which we are making a chart. 
If these constants be coefficients of the variables, we have an 
equation of the type bY =aX-+cZ, we shall have the scale 
moduli, m,=bm, and m,=am. Thescale modulus of the z-scale 
for diagonals need not be calculated, Z is much more easily 
entered upon the chart from observations of the actual values 
for various points after the co-ordinates have been calibrated. 


y =x and hence, in full y/"- | =x 


The formula bY =aX+cZ can be written PSO, 


i t a : 
this will enable us to make m,=m and Mz = M, which may 


‘For a description of the engineers’ rules or scales, see Chapter XVII, 


CURVES FOR FORMULAE 519 


give an easier plotting scale directly from the ruler. Thus if 
we have 14Y =7X +3Z, it is a convenience to plot.y =mY and 


7 
x=7qmx =3mX, for we can plot and calibrate directly from 


the 10 and 20 sides of the ruler; but if we have 5Y =2X +3Z, 


we 


or =o +.32 It 18 more convenient to plot y=}mY and 


x =3mX, for we then use the 20 and 50 sides directly. 

Of course these considerations are largely directed at the 
- simple co-efficients, but they hold also for more complicated 
ones. When we have an equation such as 157 =37.295X+Z, 
no rulers will serve directly and it would not pay us to plot 


37.295 
x=—Foq MX =.237 mX from a specially constructed scale 


(best secured by the method of triangulation?), instead we need 
only plot x =.25mX, which we can do from the 40-side of the 
rule, and shift the direction of the z-diagonals a little. When 
constants are added in the equation, the effect is not to enlarge 
or diminish the size of scales, nor to alter the scale-moduli in 
the least, but it is to shift the scale numbers, without otherwise 
disturbing them, along the axis. No matter how many con- 
stants be added, they can of course be lumped into one, thus 
bY =aX+cZ+k. Obviously the correction for the constant 
must be made upon one or another scales, that is the constant 
must be attached to one or another variable, to get it into the 
chart. Thus (bY —k) = aX +cZ, b¥ =(aX+k)+cZ, and bY = 
aX+(cZ-+k) are all forms of the same equation. The amount 
of shift is proportional to the amount of k, but care must be 
taken to divide it by the coefficient, if there be any, of the 


variable to which it is attached. Since aX +k =a(X+—), we 
k : , : 

must make x =ma(X +") and if we wish to shift the scale 

(i.e. add the constant) after the scale (i.e. x =amX) has been 

plotted we must shift it by the amount of £, Not.of Fe itis 

sometimes simpler to make the correction while plotting, that 


k : eh 
is plot for x =m(aX +h) or x =am(X+—), directly by sliding 


2 For a description of the amplifying or diminishing of scales by triangulation, see 


Chapter XVII. 


520 CHARTS AND GRAPHS 


the ruler along until the calibration X is at the point of 


(X44). 

We have so far considered only one form of calculating chart 
the distinct feature of which is the parallel straight line z- 
diagonals. This is the chart for all equations involving the 
sum or difference of two variables of the first degree. It may 
be called, therefore, the additive chart. It is by far the most 
important. and useful, as well as the simplest chart of its kind. 
Moreover, it is the basis for so many other calculating charts 
that we have dealt with it in great detail, almost all of which 
will be essential to an understanding of the other types of 
calculating charts. 


o 1 2 3 4 5 6 7 8 Ee SX) 
Fig. 403. 


The reader may note, however, that the use of the scale- 
modulus for the projection of the z-scale has been expressly 
enjoined. This is a peculiarity of the rectangular chart, the 
z-scale being best laid off by inspection in it. The reason for 
this is that the scale of the zdiagonals is not easily commen- 
surable with the x and y-scales. The diagonals form angles of 
45 degrees with the other co-ordinates, when the scale-moduli 
of the x and y-scales are precisely adjusted; they form approxi- 
mately the same angles when the adjustment is not complete 


CURVES FOR FORMULAE g21 


but is fairly close. Now if we could lay off all three sets at 
precisely equal angles the z-scale would become easily com- 
mensurable, and the scale-modulus of the z-scale could be used 
like the other scale-moduli. 

There is much to recommend such an arrangement of tri- 
linear co-ordinates. All intersections would be distinct, hence 
greater accuracy would be achieved in the use of the Chak. 
The useful portions of the chart would be more compactly 


LS SA 
ASA ae 
A SURLY. 
NTN ee 
EERIE 
A ACA 


avers oN 


SS 

Pas 

ESS KX x ANA, 
Fig. 404. 


positioned, hence space would be conserved and greater detail 
available. The form would be unusual and more attractive, 
an important feature, since, as we shall presently see, these 
charts are more pictorial and popular than business-like. Yet 
in spite of these advantages, the equilateral and equi-angular 
form is seldom or never used, probably because the average 
chart-maker has become so accustomed to rectangular co- 
ordinates. 

The additive chart which we have described can be turned 
into a factorial one by logarithmic projection of scales. It 
can then be used to calculate the formula Y = kX°Z’, since log 


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CURVES FOR FORMULAE 523 


Y=log k+a log X-++¢ log Z. The chart is prepared precisely 
as is the additive chart, and is much more generally useful. 


2 


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Fig. 406. 


An exponential chart can be obtained by the combination of 
logarithmic and arithmetical scales, for the formula Y = a*Z 
and the like. Other possible combinations will occur to the 
student. The chief use of the chart, however, has so far been 
in its additive and factorial forms, having arithmetical and 
logarithmic scale projections only. With these two the student 
should be thoroughly familiar. 

We come now to another chart which has straight-line 
z-curves or diagonals. It is easily distinguishable from the fore- 
going from the fact that the z-curves are not parallel to each 
other, but radiate from a common intersection point. The 
parallel line chart was simply a multitude of curves for the 


linear equation, Y =aX+C, in which many values of C were 


taken and C itself treated as a variable. The radiating straight- 


— 


524 CHARTS AND GRAPHS 


line chart is simply a multitude of curves for the same linear 
equation, save that many values of 4 are taken, and 4 is 


Plotting Forsule. 


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ESY TO SCALES FoR CHARTS 


Fig. 407. Chart for Determining Scales of Curve-charts. 
At the bottom of the chart find the value of the highest point in the curve. At 
the left-hand side find the height which you wish to give it on the chart. At the 
nearest intersection of the ordinate with a diagonal at this height read (in the 
scale of diagonals above and to the right) the side of the rule, N, and the plotting 
value of each unit on this scale (U). 
treated as a variable. For convenience we will still call this 
third variable Z, and the equation therefore which this chart 
solves is of the form Y = ZX, when the common point through 
which all diagonals pass is the origin of the x- and y-axes. 
When the common point is located elsewhere on the y-axis, 
as at the co-ordinates x = 0, y =c, the chart solves the equation 
Y=2 X+c. The chart is primarily factorial, though it has 


CURVES FOR FORMULAE 525 


arithmetically projected x- and y-scales. The z-scale is a scale 
of angles, a circular function of the x- and y-scales. 


AAZ 
TAM, 

VV 

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a 

Fig. 408. 


We shall not go into the principles of this chart in detail, 
it can be easily made, the x- and y-scales being arithmetical 
and the z-diagonals and z-scale being best put in by inspection. 
The chart is not of much value (save for one particular pur- 
pose), it is difficult to use accurately when the values must be 
interpolated between co-ordinates and diagonals and loses 
detail as the diagonals converge. For the processes of multi- 
plication and division which are the main purpose of this arith- 
metical factorial chart, the logarithmic factorial chart is in 
every way, save one, the more satisfactory. 

There are a great many other calculating charts upon co- 
ordinates, which have not the straight line diagonals or z-curves, 
but in which the third variable is shown literally by a series 
of curved lines, each bearing a particular value of Z. These 
are, of course, only multiple parabolic, hyperbolic, or other 


CHARTS AND GRAPHS 


Coma 


-_ 


_ curves (even including circles and ellipses), and solve more 
complicated equations than the forms already discussed. 


CURVES FOR FORMULAE 527 


100 
Gaps 
9 im: 
: aN 
ez 


w 


. 
Orror ~w Pr mM@wO 


(Se 


Fig. 411. 


They are largely of academic interest, however, forming in- 
teresting exercises for the student and somewhat highly pic- 
torial displays of the behavior of phenomena to which their 
formulae apply. For practical purposes they are of little value, 
since they take up much time and effort in the making and 
are neither so accurately nor so easily used as the calculating 
charts to which we shall later come. 

We come lastly to the multiple calculating chart, a com- 
bination of two or more simple charts of the types described. 
In the instances considered, the charts have shown only three 
variables, and therefore been suitable only to equations with 
two variables beside the root of the equation (in itself a vari- 
able but dependent upon the other two variables). More 
complicated formulae, with three, four, five or more, inde- 
pendent variables, can also be shown by these calculating 
charts, by the simple trick of joining together a number of 
individual charts. Thus the first chart can show two inde- 


528 CHARTS AND GRAPHS 


0 10 
3 r) 
8 8 
7 7 
6 6 
s 5 
2a 4 
3 3 
2 2 
’ 7 
© Oo 
7 cf 
-2 2 

-3 3 

<4 8 

-5 Ss 

<6 6 

“7 7 

-8 “8 

-9 2 

= 


“10 
ALIGNMENT CHART FOR SOLUTION OF QUADRATIC AND CUBIC EQUA TIONS. 
From “Graphical and Mechanical Computation" by Joseph Lipka, published by John Wiley & Sons, 
by permission. 


Fig. 412. A More Complicated Chart for Solving Quadratic and Cubic 
Equations, 


The presence of this chart in this chapter was discovered too late to shift it to its 
proper place in the chapter on Composite and Zigzag Nomographs, about page 576, 


CURVES FOR FORMULAE 529 


pendent variables on its x and z-scales, and their resultant 
upon its y-scale. The y-scale of this chart can be used as the 
x-scale of a second and adjoining chart, a third independent 
variable appearing on the z-scale of this second chart and the 
new resultant on its y-scale. 

By continuing to join new charts to the old ones, the number 
of variables in the equation can be increased indefinitely. 


ASSES > 
i 
Seek ec 
ee 
cis aia iS ab ha a a eae 
BSBPERERRR SESE Eee 
; Bebe RE RB RERER ER SES * 
BS SEERNESKISRERERISEEET « 
ERE 
ae eae CBRE SEER ERER ES SSE’. & 
_ RERERERSCBESEENSEERERERERER ERI,” 
est EREKEEKE RE CEESREREREKE 
won SRISRISKISKISL SISREEERSKEREK 
o -PSRERERERET Bib CEREEREKEKEK ET 
-DREKEREKEEERE SKEKEKEKEREKE 
BESET SERRE 
RSRSRSRSEERERERMREREKEEEK ER eS 
SesbsbseserRskskNeskseerpeirisskese 
RE ERES ESET RSeSE Nes RERERER SSeS 
<eseSRSerRISREENSRERERERERE SES 
REE EERE ES 
ese Rsk“ eEREREKEKEK EK esl 
FESS ERE ERE 
a re F vere : SSOEREREREBEEEEERE ie 
SRERERERERERIS 
SREREREREKISS 
SRSKISKIKISREK EL: 
reps <EBREESEER ERT 
o SReeEESepP 


Fig. 413. A Simple Model of Profit-and-Logs Computer. 


Where all the independent variables are factorial, we can use 
the logarithmic projection throughout with parallel straight- 
line z-curves on every chart, but where some of them are 
additive, it is not possible to use the logarithmic projection. 
It is for such cases that the arithmetically projected factorial 
charts last described come in handy, as by their use the addition 
processes can be performed upon additive charts and joined 


530 CHARTS AND GRAPHS 


to factorial charts through the common arithmetically pro- 
jected scales. Often, the various charts are not set side by 


PROJECTED AREA OF CROSSHEAD BEARING SURFACE IN SQUARE INCHES 
ofl Kot £6 8.2 558 3 * S15 * F 


N 


Z 


5 
1g Wy, 


N 


3 
ty AVS. 
XV SANA 


NY 
L/S 


Ms 
2 
3 8 


AE Y 
wlll YYZ 
| 


cE 7/7/ 
Lae 


{VV 


We 
KY 


\\ 
\) 
a 


LY 


TOTAL LOAD ON CROSSHEAD GUIDE IN POUNDS 


Xn 
E 


\\\ 


\ 


2 


\\ 


FAN | 
N 


PROJECTEO AREA OF MAIN BEARING IN SQUARE INCHES 


A 
iN 


vi) 
mi 


= 


388 iar eos 


= =} S ‘= c “= = 
~ 


= 2 & Wf & 
TOTAL LOAD ON PISTON IN POUNDS 


\ 


From E, A. Andrews, in ‘‘Machinery" and Haskell's ‘‘How to Make and Use Graphic Charts." 
Fig. 414. A Composite Chart With Many Scales. 
Showing Loads on important engine frame members. 


side but are superimposed, the reader being asked to follow 
a sort of mystic maze along these ambiguous co-ordinates till 
he arrives safely upon the “home’’-line scale and meets the 
answer to his problem awaiting him there. 

All of these charts are more sensational then satisfactory. 
Needless to say, their preparation consumes much time. A 
large amount of excellent zeal is sometimes displayed by. in-’ 
experienced chartists in the formation of beautifully-drawn 
and elaborate chart-forms, suitable for equations with many 
variables. It is always disappointing to observe the beautiful 
work and the great energy which has gone into the prepara- 


$31 


Cup bg sd edury - Aypnteodg JU9IN dose in 4, 
ea a ee ee KIEL pe 
& 


$ 
3 ¢ $ 8 , © KASS OS oe , 
yg % 
ae SOS 
fen red % 
LISELI OLS PERI, 
EES AE AGES SRE 


Fs ORR OK RS Ox : 
FS SSK ERR Cage 
itn < 3 one 

| LITHIA WRAL Ce 
PEE EEE EEE WCE WX es 


DE] Og s0d “Gary » Ayou0g surusnD sdag 


“How to Make 


ding the multi- 


and Haskell’s 


for the truth must be told 


le Combination of Logarithmic and 
les by the Use of a Curve. 


CURVES FOR FORMULAE 


From B..B. Hood, in “Metallurgical and Chemical Engineering,” 


and Use Graphic Charts.” 


A Simp 


Arithmetical Sca 
Showing the proper current density for copper transmission liner. 


Fig. 415. 


tion of these calculating charts, 


that both in their preparation and in their rea 


532 CHARTS AND GRAPHS 


curve equational chart forms are uselessly wasteful of time and 
energy. Everything that can be accomplished by these elab- 
orate and beautiful charts can be accomplished much more 
simply, accurately and easily by the use of the charts which 
will be described in the following chapter, in which the intri- 
cate network of co-ordinates? and curves alike is entirely omit- 
ted and the scales alone are presented upon paper utterly de- 
tached from their fields and curves. 

3 It is indeed true that the co-ordinates need not be used on the curve if rectangular 


movable axes (similar to isopleths) on separate transparent sheets be used to project 
*he co-ordinates of the point to the scales where they may be read, ; 


CHAPTER XLVI 
PARALLEL NOMOGRAPHS 


In the calculating charts just discussed, we noted that 
the values which solved a mathematical equation lay along 
a curve and that the chart was more easily constructed and 
accurately used when these curves formed straight lines. From 
this last condition, it is but a step to conclude that the straight- 
line curves themselves could be omitted and a movable 
straight-edge (ruler-edge or tightly drawn piece of thread) 
could be used in their stead, the reader of the chart being re- 
quired to adjust the straight-edge afresh for each reading. 
The only objection to this step is that while the lines are 
straight lines, their angles are arbitrarily set by the problem 
and the straight-edge must be adjusted at a certain angle or 
slope before it can be used. But in the charts which we will 
now consider, this obstruction is removed, the charts being so 
designed that interpolation by means of the straight-edge is 
possible in any and every position of the edge, the equation 
being satisfied always. The straight-line transversal is no 
longer called a curve, but is now known as an “‘isopleth,” the 
points through which it passes being always of equal value, 
that is, forming an equation. The scales are now called “axes” 
in a wholly isolated sense. The chart itself is called a ‘‘nomo- 
graph,” “nomogram,” or “alignment ch art,” the latter name 
being obviously derived from the fact that the proper corre- 
sponding values are always in perfect alignment. 

In the nomograph, the network of co-ordinates and the 
plotted curves themselves being omitted, there are three 
scales alone retained. These scales are the scales for the two 
axes of the curve and the added scale of the diagonal curves 
themselves. But the scales are so carefully arranged, both as 
to their projection and as to their position, that the intersection 
of a straight line or isopleth across two scales always gives the 
proper corresponding value upon the third scale. The arrange- 


533 


534 CHARTS AND GRAPHS 


ment of these three scales is either parallel or zigzag, so that 
nomographs can be divided into two classes, the parallel and 
the zigzag nomographs. Many! other forms are possible, but 
are largely of academic interest. The two principal forms are 
the simplest and most satisfactory for all work. 
The parallel nomographs are based upon the geometrical 
theorem of similar triangles. If we take the simplest case, in 
which the three parallel axes or scales, which may be called 


Veyry, 2 Z4Xe x + 2) 


Fig. 416. 


the x, , and 2 scales, or axes, are so arranged that the two 
outer ones (let us say the x and z axes) are equidistant from 
the middle or y-axis, then, we will see that whenever two lines 
(isopleths) cross these three axes, the distance laid off on the 
middle axis will be the average (arithmetic mean) of the two 
distances laid off on the outer axes, between these cross lines. 

To make this quite clear let us set three rulers up on end 
against the wall at equal distances along the wall. With the 
rulers resting on the floor, their zero-points or lower ends will 
be in a straight line, the line of the floor itself. Note that 
the floor here forms an isopleth, the average of the two outer 
zeros being shown by the middle zero. Now if we hold a 
piece of string tightly stretched across these rulers, we will 


Cf. Lipka, Joseph, Graphical and Mechanical Computation, John Wiley & Sons. 
Peddle, John 'B., Construction of Graphical Charts, McGraw-Hill Book Co, and 
Running, Theodore R., Empirical Formulas, John Wiley & Sons. 


PARALLEL NOMOGRAPHS $35 


see that the value on the middle ruler always equals one half 
the sum of the values on the outer rulers. If the string passes 


12 


of 
oo? 
oo? 
o? 
Phase) 


Base line 


ve a. +2) 
Fig. 417. 


across the first or x-rule at the point of six inches, and across 
the third or z-ruler at the point of ten inches from the floor, 
it will obviously cross the middle or y-ruler at the point of 8 
inches, one half of the sum of six and ten. The general form 
of the equation 1s 
x+2z . 
‘artes or 2 y=x--2. 

To adapt this device to the processes of addition and sub- 
traction is a simple process. Let us merely substitute a ruler 
calibrated to half-inches for the inch-rule in the middle. In 
other words, let us substitute for the y-scale a scale with values 
of Y such that each value of Y is just twice as large a number 
as its actual y-distance, that is, Y=2 y. Now the readings Y 
on the y-scale will be, not the average, but the sum of the 
readings on the x and z scales. The formula for the chart be- 
comes 2y=x+z, or Y=X+Z. And for all positions of the 
cross-line or isopleth, the intersected points Y, X, and Z, will 
have the relation Y=X+Z. Subtraction may obviously be 


536 CHARTS AND GRAPHS 


performed on such a chart by adjusting the straight-edge or 
isopleth through any given values on the X and Y or on the 


12 12 24 24 
23 23 
1 1 2 22 
a 1 
lo 10 20 9 ie 
19 19 Le = 
9 9 18 18 aia = 
Ug are a 
tan 6 ie aia 
a= 167-15 
7 yop — al Bila 
oP aa 134-18 
6 & oe 12——12 
ul 
5 $ 10 ) 
9 r) 
4 $ 8 8 
1 7 
3 $ 6 6 
6 5 
2—— 2 s . 
3 $ 
i-~3 shee 
i 3 
oo oe 
x ¥ % 
> oa a 
Fig. 418. 


Z and Y scales, the difference being shown on the remaining 
(or other outer) scale. For if Y=X-+Z then it is clear that 
XA’ =Y —Zand Z=Y —X. The middle or y-axis always carries 
the minuend in this arrangement, just as it always carried 
the sum when the same arrangement is used for addition. 
Before going further let us again carefully take stock of the 
algebraic symbols which we shall use in this chapter.2. The 
chart, as we have seen, uses one or more sets of three axes, 
which we shall call the (x), (y), and (z) axes. Along each 
of these axes we measure distances, x, y, and z, in terms of 
various units of length, or scale moduli,’ m,, m,, and m,, all of 
which are readily convertible into a common unit of length, 


2 See also previous chapter. 

*Throughout the formule for nomographs in this book, the scale-moduli mx, my, 
and mz, have been used to indicate the plotting instructions for the variable X, Y, 
and Z. These formulae, therefore, differ slightly from those of Professor Lipka, in 
whose book the scale-moduli are used to indicate plotting instructions for the func- 


tions of the variables, f(x), f(y), and f(z). 


PARALLEL NOMOGRAPHS $37 


or chart-modulus, m. The values which are entered or cali- 
brated at these distances are X, Y, and Z. These last, X, 


10~7—20 100——100 10 ——10 
Z 2 80 80 9 9 
8 C) ‘ : 
60 60 8 8 
ef T 60 ° 1 1 
6 6 40—1—40 , " 
$o 30 
é r) 6 F 
20 20 
4 4 ‘ ‘ 
10 10 
$ & 9 9 3 3 
8 8 
1 7 
6 6 
6 5 
2 2 4 2 z 
3 
2 
1 2 s—! 3 “t- ry 
5 4 ¥ % 
ye 
Fig. 419. 


Y, and Z, are the symbols of the variables in the equation 
plotted, they are the scale-figures which appear on the chart, 
which afford, by their readings along the isopleth, the solutions 
to the equation. 

To adapt this device of three parallel scales to the proc- 
esses of multiplication and division we need merely change 
the calibrations on the three scales to a logarithmic projection. 
As always, the actual distances, x, y, and z, on the three axes, 
still have the relation 2,y=x-+z. But we plot on the x-scale 
the values of log X, on the z-scale the values of log Z and on 
the y-scale the values of 4 log Y, using the plotting equa- 


tions, x =m log X, z=m log Z, and 2y =m log Y or yas log Y. 


Since 2 y=x-+2, log Y=log X+log Z and Y=XZ. The read- 
ings of Y are the product of the readings on X and Z, and 
multiplication is accomplished by adjusting the straight-edge or 
isopleth through given points on X and Z and reading their 


$38 CHARTS AND GRAPHS 


product on Y. Division, like subtraction, is accomplished by 
adjusting the isopleth through points in the middle and one 
outer scale, the answer being read on the other outer scale. For 


if Y=XZ, then x=4 and z=. The middle or y-axis al- 


ways carries the dividend and the product in this arrangement. 

As it will be seen that the distance on the middle scale is 
always the average of distances on the outer scales, we must. 
expect normally the resultant, that is, the sum or minuend 
(arithmetically), or the product or dividend (logarithmically) 
to appear on the central axis only. But a rearrangement of 
scales can be made when it is desired to place this variable 
on an outer axis. For this we must use complementary num- 
bers in addition and subtraction, and reciprocals in multiplica- 
tion and division. That is, we must substitute 0 —X, for X, 


in addition, and ¥ for X or 0 -Ieg X for log X in multiplica- 


tion. In this arrangement of the additive nomograph (for 
additions and subtractions) we make x=m(-—X) or —mX. 


Then since 2y =x+-, we have the equation Vad at ya 
m m 


-~X+Z=Z-X. And since Y=Z-X, then Z=Y+X and 
X=Z-—Y. In short by upsetting the scale on the first axis 
we have exchanged the meanings of the scales on the second 
and third axes and the third axis now is the resultant (sum or 
minuend). The rearrangement of the factorial nomograph 
(for multiplying and dividing) is similar. Here we make 


x =m(0 —log X), with the result that —log xy-* 4 ee 
m 


log Y —-log Z and log X =log Z —log Y and x=5 orL=AY, 


The same effect is noticeable as before, the upsetting of one 
outer scale shifting the meaning of the other two scales, the 
other outer scale becoming the resultant (product or dividend). 

Writers on nomographs are accustomed to attach import- 
ance to the position (vertically) of the scales along the axes, 
a detail to which the cases of reversed or upset scales which 
we have just considered, naturally leads us. It has been as- 
sumed in this discussion that you have kept in mind the idea 
of three rulers stood up on the floor against the wall, for this 
makes clear that the three axes must have a common base 


PARALLEL NOMOGRAPHS 539 


line, or isopleth passing through their ‘“‘zero-distances” (regard- 
less of the calibrations which may be assigned to these dis- 


~ MD O\P TH oO 
~ BND » © 


“ae Fig. 420. 


tances). Thus when the scales are reversed as in m( —lag X) = 


= Ea Te ; : ‘ 
x or log X = ——, it is obvious that we are changing the di- 
m 


rection of measurement and counting downward into or below 
the floor level and an isopleth across the three rulers would 
have to pass up through the floor. Nowit is not at all necessary 
that the base line (i.e. floor line) be at right angles to the axes 
(or rulers); our line passing through common “‘zero-distance”’ 
points or “origins” of the axes can be a very steep diagonal. 
And when one of the scales has been reversed, it is distinctly 
better to use a diagonal base-line so that the isopleths used in 
solving the problems by the formula of the chart, shall be as 
much as possible at right angles to the axes, to facilitate 
accurate readings. 


540 CHARTS AND GRAPHS 


Frequently, in fact, more often than not, the values in which 
you are interested do not begin at zero, but begin at some 
distance up the scale, that is to say, the useful or desired range 


x Y z 
Yextz 


Fig. 421. 


of the variables does not come down all the way to the base- 
line or origin of the axes. In that case again, it 1s well to use a 


10+ 10 38 
37 


38 66-58 


36-+36 67-757 
38-736 
uy 34434 5e-+56 
3$+s3 
52-782 65-56 
31431 
12-12 30-730 64-64 
29-+29 
28 28 63-++63 
27-127 
13-13 ' 26 +26 s2-+62 
25 +26 
24 +26 61+61 
23 +23 
14+14 22 +22 60-+60 
x Y Zz 
Y22-2 
Fig. 422. 


diagonal base-line, in order that you may omit the lower parts 
of the scale entirely, together with the base-line itself, on your 


PARALLEL NOMOGRAPHS 541 


finished chart. Another way to achieve the same result is to 
alter the calibrations alone, so that x=m(X+a), y=im 
(Y +b) and z=m(Z-+c), (in which a, b, and c are constants 
which in themselves satisfy the formula). In general, the 
values of these constants should be such that they are equiv- 
alent to the lower limits of the desired ranges of the variables. 
The real object of the diagonal base-line or diagonal zero 
isopleth, is to make all isopleths which will be used on the chart 
as perpendicular to the scales as possible. The nearer to a 
right-angle the intersection of isopleth and axis becomes, the 
more sharply the two lines cut each other and the more easily 
will accurate readings be made. 


This brings us to the important element of the range of 
the variables. For it is not necessary, nor even possible, to 
picture all the possible values of a variable upon a chart. In 
actual problems the independent variables will usually be found 
to fluctuate between certain limits. It is thus unnecessary to 
use a scale so great that it shows values in excess of the maxi- 
mum limits, or to include on the scale the values below the 
minimum ‘limits. Space is conserved and detail gained by 
making the range of the scale conform to the range of the 
useful values of the variable. And when the ranges of each of 
the two independent variables have been set, it is easy to find 
the range of the resultant or dependent variable (the root of 
the equation). In the previous chapter we have indicated a 
method of noting these limits, thus* 

yer|!9] + 2|ghorr|'3| = |'2] +215) 

A variety of methods are at hand for confining the chart to 
these ranges. We may place the lower limits at the zero dis- 
tances or origins of each axis. Or we may place the maxima 
upon the level (or base-line). Or best of all, we may place the 
mid-points along each range upon a level isopleth. The 
advantage of the last method is that all the possible isopleths 
will then cross the axes at angles nearer to a right angle than 
by any other arrangement. Having approximately positioned 
our scales with this object in view we do not actually need 
to calculate the values of the mid-points (fractional as these 
may be), for plotting; we need only calculate the values for 
any round numbers and precisely position the scales about 
them. 


542 CHARTS AND GRAPHS 


An important point in the making of the chart is its total 
size and proportions. Both its height and width should be 
great enough to serve whatever purposes of convenience in 
use, legibility and detail of readings, visibility at certain 
distances, or success in reproduction and reduction, will natur- 
ally obtain in chart-making, but the width should always be 
at least as great and if possible half as great again as the height. 
If the chart is too narrow many of its useful isopleths will cross 
the axes at such small angles that correct readings are difficult. 
If the chart is too wide, the isopleths will all cross at very good 
angles but the scales will be so closely compressed as to make 
detailed readings hard. The best form in general is one in 
which the most steeply sloping isopleths cannot cross the axes 
at smaller angles than from 45 degrees to 60 degrees. The 
height should be approximately two-thirds the width. 


The final consideration is the choice of axis for the de- 
pendent variable. By the dependent variable is meant the 
variable whose values are sought from given values of the 


fiapa 100 ——100 ap etO 
20 + 80 Galas 
60 60 8 8 
50 50 7 ? 
40 40 
6—+ 6 
30 30 
2 6-5 
20 + 20 
4 4 
3 
lo 10 
3+ 3 
4 6 8 
6 6 
5 4 4 2 2 
Ps 3 3 
7 2-+ 2 
8 
9 
0 
aos sti 
x 2 z 
2 
S es 
? x 


Fig. 423. The Inverted X-Scale. 


PARALLEL NOMOGRAPHS 543 


other variables. Of course it often happens that the same 
equation is often used backwards, and that at times one vari- 
able is sought from given values of the other two and at times 
another is sought. But usually there is one variable which is 
most likely to be the unknown and this should be treated as 
the dependent variable. The best axis for the dependent 
variable is always, ceteris equibus, the (y) axis. For then all 
needed isopleths will lie within the limits of the two outer 
scales and the three scales can be of roughly uniform height. 


10 


7 3 
/ 
fa 
Va 
7 2 
—10 
§ ASS 
Y 8 oN 
ve 7 ~ 
Zo af Ae 
5 5 SS 
~ 
4 4 ~ 
aay 
3 3 Bars 
ie 
, pie oS .9 
a 
2 2 ys. a) 18 
\ - ‘ 27 Aen 
\. Pkg i 
YSN 
\ 
6 
bale 
-* 4 
x 4 ; 
\ 
\ 
\ cu 
& . 
Y 
= .N 
CHEN 


Fig. 424. The Use of an Outer Scale for the Unknown 
Variable is Not Good. 


544 CHARTS AND GRAPHS 


Were the known variable placed upon an outer axis, it is 
clear that it would have to-be extending above and below 
the levels of the other axes unil it included the most ex- 
tremely sloping isopleths which could be drawn through the 
central and other outer scales. The result would be a chart 
of very irregular appearance, wasteful in space and involving 
less accurate readings because of smaller angles of intersection 
between isopleth and axis. The danger of errors in placing 
the isopleth would be four times as great, since the errors in 
positioning the known values may be doubled upon the un- 
known scale, whereas they are halved when the unknown scale 
is on the central axis. 


It has been the purpose of the foregoing discussion to be 
suggestive rather than definitive, of the general principles of 
the parallel nomograph. It remains to examine this chart 
analytically. This will lead us at once to a generalized form 
of the parallel nomograph, with important modifications which 
make it far more flexible, in use. The chart has so far been 
considered only with equidistant axes. 


The geometrical proposition of similar triangles can equally 
be applied to axes which are not equidistant. When the inter- 
val between axes (x) and (y) is equal to that between (y) and 
(z), then the formula for actual distances is, as we have seen, 


zyx 


At OF 2 y=x-+z. And if we denote the total distance 


Veo 


(pray =o% tP> 


Roe eae 
¥* peg? they” 


<----p----3<--—-- See Se = 
Fig. 425. 


PARALLEL NOMOGRAPHS Gas 


between the (x) and (z) axes, either measured perpendicularly 
to the axes or along the base-line or along any isopleth, as 
“p+q,” taking “p” as the part between the («) and (y) axes 
and “gq” as the part between the (y) and (z) axes, we may write 
the formula for the distances or axes cut between isopleths as 


alone 3 labatinlaand ag 
(p + 4) (p + q) pt+q 
This formula is applicable to any parallel nomograph, no 
matter at what distances from each other the axes may be 
placed. And the significance of “‘p” and ‘‘q’” are very easily 
seen. ‘They are the coefficients of the x and z variables in the 
additive formula 2 Y = X + Z, and the corresponding expo- 
nents in the factorial formula Y2 = XZ, becoming coefficients 
in the corresponding equation 2 log Y = log X + log Z. Inshort 
the complete formulae are, for additive charts (p + q) y = qx + 
pz, and for factorial charts, y “°+® = xz”. Obviously when 
p and g are equal they may be written as 1 so that we have 
2y =1x+12 and y? = x! 21. So that the formula at once 
explains the half-size scales taken for the middle axes. Also 
when we reversed or upset one scale, we were in the additive 
formula inserting a — | coefficient, making the value of g = — 1. 
We were then obliged to shift the other outer axis into 
position midway between the first and second axes (a process 
which we spoke of as exchanging meanings of scales) so that 
p became + 2 and (p + q) became + 1, so that the formula! 
became 1 y = —1x+2x. Thus the general formula (p + g) y 
= gx + pz covers all cases of the parallel nomographs. 
We are now ready to lay down the rules for the construc- 
tion of the parallel nomograph. In the first place, we have an 


y ,or (p +9) y = qx + pz. 


4 Or, calling the y-axis z, because it is now the third, and the z-axis y, because it is 
now the second, we have + Iz = — lx + 2y, which agrees exactly with the formula for 
reversed scales. So doing, we maintain the symbols (x), (y) and (z) for the axes strictly 
in the order in which these axes appear on the chart. 

It is obviously better to permit the symbols (x), (y), and (z) to adhere to the axes 
wherever they appear, regardless of their order upon the page, as the general formula 
then applies consistently and without confusion. In the text from this point on, this 
has been done, and the (x)-axis need no longer be the first, the (y)-axis the second, 
nor the (z)-axis the third; but the algebraic signs of p and q will signify changes in 
position, and the algebraic signs of the scale-moduli will be significant of the direction 
of plotting. 


546 CHARTS AND GRAPHS 


#54+CZ # + K, which’ 
L,) , IL, 

we wish to present upon a chart or diagram which has the 
relations of gx + pz =(p + 9)y. We give this diagram any 
height, 7m, we wish and approximately half as much more 
width. If, as is most convenient, we let the chart-modulus, 
m, equal 1 inch, then 7 is the total height of the chart, or 
length of each scale, in inches. Now along these scales we 
propose to plot the values of the independent variables, X 
and Z, from their lowest, L, to their highest, H, useful values. 
Call the difference between these extremes the range, R, of 
the variable, then 

R,=H, -L, R, =H, -L, 

We can easily plot the values X and Y through these 
ranges in these given lengths by the method of triangulation 
if, as is generally the case, the intervals are not even fractions 
of the inch.6 Then draw an isopleth through any convenient 
values of X and Z on the (x) and (z) scales and we know that 
the corresponding value of Y lies somewhere along this iso- 
pleth. Substitute these values of X and Z in the equation and 
learn the corresponding value of Y. Select a second conveni- 
ent value for X and substituting it and the Y-value for X and 
Y in the equation, solve and get a second corresponding value 
of Z. Draw a second isopleth through the second values for X 
and Z on the (x) and (z) scales and since the value of Y has 
remained unchanged, we know that the (y) axis passes through 
the intersection of the two trial isopleths, and is parallel to the 
other axes. Now solve a few more equations containing con- 
venient values of X and Y and draw their isopleths and you 
will rapidly calibrate the (y) axis with its Y values. After a 
few points have been plotted the rest of the Y-scale can be 
put in by a ruler, and the method of triangulation. If these 
directions are carried out the entire chart will be finished in a 
short time. 

The student will look however, for an analytical method 
which will define mathematically the various scales and their 


equation of the general type Y= 4X 


6 Within the short vertical parallel lines in the equation are inserted the high, H, 
and low, L, values of the variables which will be required. These maxima and minima 
of the ranges are merely memoranda which do not affect the equation in the least, 
and can be omitted from the equation and noted elsewhere, if they render the equation 
confusing. 

* For the precise adjustment of scales to given sizes by the method of triangula- 
tion, see Chapter XVII, page 185. 


——e 


@ARALLEL WOMOGRAPH: 


GS9tonuveur oP svusors: 
TO INDEPENDENT VoRtaBLEs: 


TO OEPEUDENT vanragie: 


Riwits OF USEFUL veRraTIoNs: 


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25 Se) 34 
60 Ajo Werle 18 


EbLCusations FOR Yoocare: 


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veo 4 zee 16 
Te 25 2° 13 
Ye 118 Ze 4 
wr 
ye 115 
40 110 
106 
36 100 
96 
30° 90 
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25 80 
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20 10 
65 
16 60 
65 
10 50 
46 
: 40 
35 
9 30 
06 6 
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Y © (3/2/0 © (6/2)30 © 60 
xX © (2/3)50 © (6/8)18 © 16 

3X © (2/3)45 = (6/3)15 = & 
X © (2/3)25 © (5/3)13 © -§ 
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Fig. 426. Construction of the Parallel Nomograph—lI. 
Finding the unknown scale by trial isopleths. 


548 CHARTS AND GRAPHS 


positions. For this we-must know the scale-moduli, m., and 
Mz as distinct from the chart-modulus, m. The scale-modulus 
is the interval or unit distance along the scale between the 
unit values of the variables, X or Z, and is obviously written 


——— m —— 
™s ae m oa m 
or letting m =1 inch) 
2 —inche | m, =— inche 
Mx R inches : Ti ice 


Now in plotting it is always a convenience to adopt units 
of length such that they can be laid off from an engineer’s 
hexagonal rule and do not have to be specially projected by 
triangulation. The engineer’s rule divides the inch into 1, 2, 
3,4, 5, or 6 parts and decimals multiple or submultiple thereof, 
such as 10, 20, 400, 3000, etc. If we describe the scale moduli 
by the numbers of them which go to make up the chart modu- 
lus, m, and call this number S, then 


S,m,=m Sm,=™m 
and a oy 
Mm, =— Mm, =— 
SS, S; 
or, letting m =1, (that is, one inch) 
1 yew | 
My =— mM, =— 
ee Se cS: 


a 


Thus as we have previously seen, the reciprocal of the scale 
modulus, when m =1 inch, is the number of intervals per inch 
and serves as an index of the proper side of the engineer’s rule 
to use in plotting. Combining the two equations for m, we 
can eliminate it and keep measurements in terms of its recipro- 
cal, S, thus 


ee mean 
Ss, Re iS) SiR 

or 
Re Re 
S.-H S.=— 


If S in these equations becomes either 1, 2, 3, 4, 5, or 6, 
or any multiple of any power of ten, we can of course 


PARALLEL NOMOGRAPHS 549 


plot the (x) and (y) scales directly from the engineer’s rule. 
If it does not do so at once (and it usually doesn’t) always 
make it do so by altering the values of R and T slightly; that 
is, change the length of the scale, 7, or increase its range, R, 
or do both. Slight alterations of this kind do not appreciably 
affect the size or usefulness of the chart. We shall write these 
altered values in small letters, thus 


S,== S,=4 
be t, 

So we see that by selecting values for r and t which are 
close to the original values R and 7, but which make S pre- 
cisely indicate a side of the engineer’s rule, we make the chart 
even simpler to draw. 

Now we have seen that the chart has the linear relations, 
gx+pz=(p+q)y. If the chart is to express the equation, 
Y=AX+CZ+K, and we decide (since it is easiest) to correct 
for the added constant K along the (y) scale, then we write 


AX EGL oak 
and the chart must have the values 


gx=m (AX) pz=m (CZ) (p+¢)y=m (Y —K) 
eK = mZ, y =—— m(Y -K) 
q Pp 


In this the distances, x, y, and xz, are taken fiom origins 
which will be real if L,, L,, and Ly are zero, but imaginary 
if L,, L,, and L, are other than zero and the zero or base-line 
ig not shown on the chart. Also we know that by definition 


x= My 2=m,L y=my,(Y —K) 
Hence 
A m —~m My = : m 
M,=— m 1, =— = 
a q Pg 


But from above 


m pe 

ee M,= S 

Hence sabe nee 
iSite S. 


550 CHARTS AND GRAPHS 


PARALLEL NOMOGRAPH: EQUATION: 
Y=4X+CZ+K 
2u = 30 + 5w 
SYMBOLS, LIMITS, AND DEPENDENCE: 
A, 
+ CZ 


—5 


SIZE OF CHART (HEIGHT): LET T = 4 inches 
RANGE OF VARIATION: 
Ry = Hy — Ly = 50 R, = H,-1, =8 


RULERS FOR PLOTTING: J 
Sx = 1% | by Sz= 12/tz 
50 /4 = 12.5 8/4=2 
50° /2.5 = 20 af, 
50 /5 = 10 


INTER-AXIAL DISTANCES: 
q = AS, = (3/2) 10 = 15 p = CS, = (5/2)2 =5 


45 is 


20 70 16 


16 60 15 


13 


a 
o 


12 


o 
oO 


8 u 


v 
Fig. 427. Construction of the Parallel Nomograph—II. 
Finding rulers with which to plot the known variables and a formula with which 
to position the unknown axis. (The ruler values adopted are underlined in the 
worksheet above.) The unknown variable is still plotted by trial isopleths. 


PARALLEL NOMOGRAPHS 551 

Here we have a convenient formula for the distances, 7 
and q, between the three axes, (x), (y), and (z). Having found 
from the range and total length of the scales the convenient 


sides of the rule to use in plotting them (S =1) we merely 


multiply these by the coefficients, 4 and C, of the variables in 
the equation to get the horizontal distances between axes. 
You will notice that the chart-modulus, m, has cancelled out 
of the equations, so that p and g can be measured in any units 
which will make their sum be the desired width of the chart. 
These devices have obviated the first two trial solutions of the 
formula for the purpose of locating the (y) axis. 

Lastly we come to the formula for the plotting of the (y) 


A m m 
scale itself. Just as we wrote My =e and mM, =< SO we can 
x z 


write My = and our object will be to find S,, such that it 
¥ 


too can be plotted directly from an engineer’s rule. Above, 


: 7 
we see that m,=——m, hence 


P+q 
Be S244 245-408 
Oy PRG $ q % ‘i 


A and C, of course, are fixed, and you will often find it im- 
possible to so adjust S, and S, that while they indicate sides 
of the engineer’s rule, S, does the same. It is generally neces- 
sary, to go back to the original elements of S, and S,, namely 


"* and oh and alter one or both of them until the desired 


tx te 


result is achieved. A convenient plan is to set down columns 
for each of the elements, f,, 7x; Sis 95 99s Ps. 9m Te ANG ty 380 
that you can try a number of different values of 7 and R, 


for each variable before you give up hope. 
When S, cannot be made to conform to a side of the rule, 


it is still always easy to project specially by the method of 


; m 
7Since obviously my =~ 
y 


$52 CHARTS AND GRA ES * 


PARALLEL NOMOGRAPH: EQUATION: 2u 2 80 + bw 
Hy Hz 
AX # C2 =» Ye 
Ly Lz 
45 20 
aa) o 5 2 » fF oO. 
“ <5 2 12 


PLOTTENG INSTRUCTIONS: 


ty Byory Sy q Sy =P 8, ry ty 
a ie ASy 9*P CBy, Se 
T26  Ry*50 Aes O-4 Ro°8 = T#6 
1 1 
6 50 8 13- 16- 3 1l- g 6 
6 50 10 16 20 5 2 8 4 
2.5 50 20 30 8756) 786 8 eo 3 
25 50 _20 30 40 10 yee 8 2 
8.5 typ | 6 9 11.6 2.5 P| 8 8 
10 50 5 7,5 9 1.5 6 9 16 
10 SOT EON IPSS: GOONER IE Fae 8 8 


Fig. 428. Construction of the Parallel Nomograph—III. 


Finding ruler-values with which to plot the unknown variable. (Any of the sets 
of underlined rulers for the three variables can be used, according to the size, 
Tx and Ts, which is desired for the chart.) Worksheet only is shown here. 


triangulation. There is no more need to make trial isopleths 
for specially computed values of the variables, and we would 
only make two or three of these when the chart is completed 
to check it up for accuracy. The whole problem of charting 


the equation Y=4X i +CZ " +K is reduced to the follow- 
ing simple steps: 
if, ee 
An CZF =Y-K 


PARALLEL NOMOGRAPHS 553. 


Ts ‘ 
Sy = is when r, approximates R, =H, —L 


Zi 
S=5 when r, approximates R, =H, —L,. 


p=CS,, q=AS,, and S,=CS,+AS,,. 
We know 7, the height which we wish to give the chart 
and which ¢, and t, approximate, and we know the width 


which we wish to give the chart, approximately which will 


be divided by the axes in the ratio of p and g. It only remains 
for us to select mid-points in the ranges of X and Z and place 
them, with the corresponding value of Y, upon a common 
horizontal isopleth and plot the scales about them. 

When the equation is factorial instead of additive, it has 


the form 
a H* A jek A 
v-(x|7|) (2iz)ex 
or log Y=A log X ake +C log Z Hens! slog K 
log x log z 


and the same treatment may be followed precisely. It is 
more convenient, however, to plot directly from a log rule, 
if one is handy, than from an engineer’s rule and a table of 
logs. We therefore drop the engineer’s rule and use the cali- 
brations on a slide-rule, if one is available. The simpler slide- 
rules have two scales, one a single and the other a double deck, 
in the length of 25 centimeters. Better slide-rules have also 
a three-deck scale. Taking the modulus, m, of the chart as 
25 centimeters instead of 1 inch (it does not make any differ- 
ence in the planning equations just listed since m has been 
cancelled out of them) we now measure 7, ¢*, and /’, in units 
of 25 centimeters, roughly 10 inches, and take S = 1 to indi- 
cate the single, S = 2 the double, and S = 3 the triple deck 
scales. In short, when S,, S,, or S, can be made equal to 
1, 2, or 3, we can plot scales directly from the slide-rule. 
Complicated formulae cannot often be made to yield 
direct ruler-copying values of all three, S,, S,, and S,, at the 
same time, either for the additive or the factorial charts. 
This difficulty is most frequently encountered in the factorial 
charts because of the more limited number of different rulers. 
When this is the case, the method of parallel triangulation 
can, of course, be used for all other values of S,, S,, and S,. 
But most convenient of all is a set of radiating triangulation 


554 CHARTS AND GRAPHS 


sheets, such as are included in Professor Lipka’ s book,’ which 
can be folded at any value of m and will give all possible pro- 
jections of the arithmetic or logarithmic scales. When such 
devices are used the chart-maker has no occasion to seek 
certain values of S, but can work with any scale-moduli what- 
ever, so that his equations become (when m, the chart-modulus, 
is 1 inch) 2 


1 


P+ 
and he can plot directly from his sheet of scales, folded at the 
proper scale-modulus. 
The general equations which have been given, namely, 


My = 


| 
y=AXe 4007" 4K 
A { 
ae r(x) (z ray K 


are usually found in simplified form, K being O in the 
additive (first) form, or 1 in the factorial (second) form. 
When a coefficient (in the additive) or exponent (in the fac- 
torial) 1s attached to the dependent variable, Y, it can be 
transferred to the other variables so as to clear Y, by division 
or involution. When the signs of either X or Z are negative, 
the sign must be treated as part of the coefficient and so trans- 
ferred to the scale-modulus or ruler-index (S), to indicate . 
that all values are plotted downward instead of upward. 

Variations of this parallel nomograph will occur to the 
student, such as charts for the equation 

VaR" 75K ape 
which must be turned into 
log. (Y —D) -log K =A log X+C log Z 


*To the chart-maker, Professor Lipka’s book Graphical and Mechanical Com- 
putation is well worth its cost, if for no other reason than for the useful scales it 
contains in a pocket in the rear cover of the book. These scales carry radiating lines 
from a common center to all parts of a ten-inch uniform and a ten-inch logarithmic 
(single-deck) scale. By folding these sheets appropriately, these scales can be obtained 
from the radiating lines at any desired smaller scale. They amount to complete 
outfits for the triangulation method of scale adjustment. 

* As is the case with Lipka’s chart-formulae. 


PARALLEL NOMOGRAPHS 555 


PARALLEL NOMOGRAPH: equation: 27.2 9 © 13.7 8? 11 ¢? 
Hy 4, ~ 4 
ax * 2 eve lest i 
ty ly 
26 50 
2 log 2 +3 log 8 = log? « tog 23:2,2,1. 
+25 06 wd 3 


* log ? + log & 1? 


CHARTING INSTRUCTIONS: 


Te 6 inches 


Ry © log 88 + log .25 = log 100 Rp = bog 50 = log .05 © log 1000 
= 2 323 

By ° T/Ry * 6/2 = 3 inches a7 ° T/hz ° 6/3 = 2 inches 

@ 7 A/a, © 2/8 © .667 p ° C/ag = 3/2 © 1.6 


By ° 1/(p+q) = 3 = .461 Inches 


20 


15 


7) 


eA DR AMDO 


R 
Fig. 429. Construction of the Factorial Parallel Nomograph. 
Finding slide-ruler plotting scales for all variables and locating the unknown 
variable axis, all by formula. (Only the adopted values are shown in the work- 
sheet, but a columnar form similar to that used in the last figure, is useful to com- 


pare different values before selection.) 


CHARTS AND GRAPHS 


556 


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eTAASPLO Weyverteyy 


PTAVSPTO 3209S FOTN 


f Type. 


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Ss 


ining 


Chart for Determ 


Fig. 430. 


Cee 
SB 8 
ar 
BGS 
— 3°) 
pu Dn 
asa 
my 
abs 
2B 
Roo 
oh E re 
Bad 
eo 
(3) i 
o6& 
ete 
LE zal 
den 
tel aleoy 7) 
ea 
n 
= OS 
oS .8 
gre 
eat 
i= 
ease 
os 
oe 
7.08 
wee 
Go lae4| 
aes 
ae 
ore 
ss) 6) fe} 
Eo y 
3 0 
o> Ss 
oe yD 
Lo ‘E 
es 
aoy 
2 
a ae 
Eee 
Y eS 
aot 
ap 
ts 
eee 
A D3 


would be useless. 


PARALLEL NOMOGRAPHS SO 


and involve therefore a special logarithmic projection with 
shifted zero. ‘The exponential equation 
. Yen tek 
can be turned into 
log Y -log K =(log 4) X + (log C) Z 
and involves a mixture of log and arithmetical scales, the 
scales of (x) and (z) being arithmetical. So does the equation 
Y =A*Z°K. 
A log-log projection is called for by the equation 
Views <2 
which turns first into 
| log Y =(log 4) +CZ (log X) 
and then into 
log (log Y -log 4) =log C+log Z+log-log X. 
The projection of powers, roots, and reciprocals all find occa- 
sional use. Indeed any function whatever of two variables 


La $8-60 @ U- 1. 
¥-10 @ U-10 


6 ("Stendere”) 


K-30 ® U- 0.1 
ZN-50 ® U- 0.1 
N-20 ® U- 0.1 


= 
> 
° 
@ 
a 
° 
- 
Height of Ordinate 
~ 
° 
Height of Field 


Value to be Plotted 
Plotting Formule 


2N-30 ® Us 0.1 
2 4N-60 ® U- 0.01 
N-10 ® U- 0.1 
#N-40 ® v- 0.01 
N-60 ® U- 0.01 
N-50 ® U- 0,01 
# N-40 ® U- 0.02 


N-30 @ U- 0.01 


1 


Fig. 431. Chart for Determining Scales of Curve-charts. 
On the left-hand scale find the greatest value in the series and on the right-hand 
scale (inside) find the height at which it is to be plotted or (outside) the height 


of chart-paper. The nearest circle on the central scale, to a straight line between 


these two will give the side of the ruler (V) and the value of the ruler unit (U). 
Comp. with Fig. 407. 


558 CHARTS AND GRAPHS 


ve One Sixteenth 
16 
4 

5 - Three Thirty-seconds 13 
faa 12 
yf One Bighth cK 
32 ie 
Y. Three Sixteentha 9 
64 ; 

Y. One Quartar-Tone 8 

8 

Kf 

5 Three Eighths ‘ 
ie 
3 One Half-Tone 
ae 5 
he Three Quarter-Tones 

WA One Whole Tone ee 4 

4 — 
i Three Half-Tones 

16 ' 


4 
Ss Two Whole Tones beep 
8 — 


7 
yi Three Whole Tones 


6 
vA 2 
2 


Distance of Variations Distance of 
Hole from of Pitch Needle from 
True Center of Music Center of Disc 

of Spiral Grooves (in tones) (True Center 
(in {nohee) of Spiral Orooves) 


in inches) 


EFFECT OF OFF-CENTSR HOLES IN PHONOGRAPH RECORDS ON PITCH OF MUSIC 
Fig. 432. Parallel Nomograph Not Chartable by Formula. 


The unknown variable, 7 ( = change in musical pitch, in tones of the chromatic 
scale) cannot be plotted by the side of a slide-rule, or any other ruler. The formula 
of the chart is, 

R+2D 


R 


in which R is the radius or distance of the needle from the true center of the disc 
(in inches) and D is the displacement of the hole therefrom (in inches). The 
formula can be stated as 


= 27/6 


a = 2 T/6 = 1 
R 5 
or log (2D) — log R = log (27/8 — 1) 
or log D — log R = log (7 Pe 1) — .3010 


If we lay off logarithmic scales of D and R through the desired ranges and then 
compute each value of these for each value of T which it is desired to plot, we 
can plot same by isopleths through the computed values of D and R. Thus: 


log 2 log 2 
T antilog (7 ) RO? 
6 6 4 PA ap or 2D D 
or .05017 T or 2 7/6 
en 050174 anne Se aes deel a ee 
a b c d e f g 
1 05017 1: 1205) erage ios Gls er 180805 
2 10034 1.2599 2599 5. | 1.2995 | 64975 
\“% 02508 1.05946 0894601 5 2973 | (14865 
etc. etc, | etc? etc. ete CEC. etc. 


PARALLEL NOMOGRAPHS 559 


can be plotted if it can be reduced to the additive form gx+ 
pz=(p+4q) y- 

The great difference between the nomograph and the 
curves described in the previous chapter, is that the curve of 
the latter has shrunk to a point in the nomograph, and the 
succession of curves has shrunk to a succession of points or 
single line. Incidentally the nomograph has shaken off the 
network of co-ordinates, though these are not essential even 
to the curve. In both of these steps the nomograph has re- 
duced the labor of chart-making and increased the ease and 
accuracy of chart-reading. We have so far considered only 
the simple parallel nomograph, which is analogous to the 
simple parallel curves, but there are other nomographs which 
serve the purposes of the radiating curves and composite curve 
charts which we shall take up in the next chapter, 


CHAPTER XLVII 
ZIGZAG AND COMPOSITE NOMOGRAPHS 


In a curious way the zigzag form of nomograph is even 
simpler than the parallel form. The parallel form has two 
outer axes and an inner axis which is slid along the base-line 
back and forth between the two outer axes as the scale moduli 
or coefficients of the variables on the outer axes are changed. 
In the zigzag form this central axis shrinks to a point—its 
own zero-point or origin on the base-line,—and having so 
dwindled moves back and forth along that base-line as a 
variable along a scale. It shrinks to a point because the third 
variable is turned into a constant. It lies upon the base-line 
because the new constant has been corrected for on an outer 
axis and one of them plotted reciprocally, that is, downward 
from the base-line. It moves back and forth along the base- 
line because the coefficient of the remaining independent vari- 
able has been turned into a new variable and hence has vari- 
able values. As a result, we must calibrate the base-line itself 
for the values of this new variable coefficient, or factorial vari- 
able, and lo and behold, we have a factorial chart without 
log projections, in many ways similar to the factorial radiating 
curve-chart for calculating formulae. 

It is simplest, however, to explain the zigzag nomograph 
independently and from a different form of the geometrical 
theorem of similar triangles. We shall now speak of the base- 
line as an axis in itself, since it is calibrated, but it is to be 
understood that distances are not measured off upon it in units 
necessarily commensurable with the distances upon the other 
axes. But we anticipate. 

If you lay off three lines or axes such that two are parallel 
and the third cuts through both, like the letter N, you can 
easily prove that along the Tee axes the Getanee cut off 
by a straight intersecting cross-line (in the finished chart, an 
isopleth) will have certain definite relations. In this case 

560 


airs 


ZIGZAG AND COMPOSITE NOMOGRAPHS 561 


we measure the distances from the intersections of the axes, 
that is, the two intersection points of the axes, are the origins 
of the axes. Let us call the first axis as before the x-axis and 


Fig. 433. 


the distance laid off on it by the isopleth from the intersection 
of (x) and (y), as before, x. Let us call the middle or diagonal 
axis, the y-axis, measuring the distance, y, laid off on it by the 
isopleth, from the intersection of (x) and (y). Let us call the 
third axis, as before, the z-axis, measuring the distance, z, laid 
off by the isopleth, from the origin of the z-axis, that is, the 
intersection of the y and z axes. Now if we indicate the entire 
length of the y-axis, from x-origin to z-origin by Q to indicate 
that it is constant, we can quickly, from similar triangles, 
verify the following statement: 
Be MS 
bun Oey 
Obviously if (with a chart-modulus, m=1 inch) we plot 


the values ~=—X, 2=Z, and—— = Y, we may use this chart 


Oy 
xX : ; 
for calculations of the equation Y =F 1 X=) 7. “Asan the 


parallel nomograph, we note that space is conserved and accu- 


562 CHARTS AND GRAPHS 


racy gained by plotting the dependent variable upon the 
central or (y) axis. 


The generalized equation 1s Y= +E. 


CZ i 


As in the last chapter, we concern ourselves first with the 
two outer scales. “Their ranges are: 


R,=H, a) 2 ae . R,=H, -L, 


If we determine upon the height of the chart, Tm, or T inches, 
when the chart-modulus is 1 inch, then the scale-moduli are 
found from the two measures of the length of the scales 


Rim, =Tm Rim, =Tm 
_Tm ee 
a eee: 


or, letting m=1, 


m, => 

a R, 
These are sufficient plotting instructions for the two outer 
scales if we are using a radiating scale-sheet.! But if we are 
using an engineer’s rule, we shall want them turned into 
values of S (the number of scale-moduli per inch): 


Sy cra . S, = s 

t, te 
and if S does not at once show a ruler-copying value, that is, 
1, 2, 3, 4, 5, 6, or decimal multiple or submultiple thereof, we 

ai oles Rand 7 a bit until it does for each scale. 

Now we can at once lay off the central or y-axis. Care 
must be taken to direct it at the true origins of the outer axes, 
which will differ from the apparent origins by the amount of 


B : : 
Gq (on the x-axis) and a (on the z-axis). The entire chart 


C 


should be about square in outline (unlike the parallel nomo- 
graph) or even slightly narrower, to get the best results in 


1 That is, a sheet facilitating scale adjustment by triangulation. These shects 
are found in Professor Lipka’s book and are described in the previous chapter. 


ZIGZAG AND COMPOSITE NOMOGRAPHS 563 


reading from its isopleths. The scale for Y can be inserted 
by solving the equation for different values of the variables 
and plotting them on the y-axis by isopleths. The scale is 
not uniform, it is a variety of reciprocal projection, each 
point having a value proportional to the ratio of the segments 
of the line on either side of the point. Every value calibrated 
on the scale must therefore be individually computed and 
plotted by isopleths. This is a simple way to construct a 
zigzag nomograph. 

Since, however, the projections through these points or 
values from any point, 7, on the z-scale, forms as we know, 
an arithmetical scale on the x-axis, it is a simple matter to 
reverse the process and plot a temporary working scale (which 
we may call W) arithmetically along this x-axis, calibrated 
equal to Y and such that from it we may easily project the 
Y-scale on the y-axis.2 We select any convenient point, 1, 
preferably near the middle, on the z-axis. Through two known 
points already computed and plotted on the central scale, we 
project isopleths from n to the x-axis. Then we lay off a 
complete scale about these two points and with n as center, 
plot their projections on the y-scale; lastly we erase the tem- 
porary scale and the point x. This is a better way to con- 
struct a zigzag nomograph. 

The student will seek a mathematical expression for the 
plotting of this eccentric y-scale. Now to find the values of y 


his! Ses 
Cy 

values of x and z. These are the distances along the two axes 
from the true origins, which differ from the apparent origins 


in the equation —= , we must first examine the true 
2 


D 
by the amounts of 5 and CG So to be quite correct we must 


write 


B D 
x=mAX + ) z=m(Z +7) 


2 The projections of Y upon the x-axis from any point in the z-axis are always 
regular (i.e., uniform or arithmetical, or in accordance with the scale of X), because 
X itself is regularly laid off, and by the formula, Y varies directly with X when Z 
is taken constant. The temporary working-scale, 7, cannot be plotted upon the 
z-axis from a point on the y-axis, because, by the formula, Y' varies inversely with Z 
when Y is taken constant; hence, the transversals from uniform intervals along the 
z-axis would only project upon the y-axis a sort of reciprocal scale projection thereof. 


564 CHARTS AND GRAPHS 


This is true from the definitions of m,and m, as scale moduli. 


So 


Mz _ mM, 
x= (AX +B) ae (CZ+D) 
Substituting these in the equation ee we have 
CANE 
O-y ~ Am(CZ+D) 
Now one = Y —E, so we may write 
Geko Ges Cm,(Y —E) 
Q -y si! Pi Sala ee Am, 
oa Cm,(Y —£) 


Q Cm,(Y -—E)+Am, 
AY Gm Y =£) 


___Cmi(¥-E)__ 
~ Cm,(Y —E) +Am, 


This is a cumbersome expression and shows that it is simpler 


to compute the y-scale empirically as before described. If we 
write 


= 


y=m,(Y -E) 
then we see that m,, the modulus for the y-scale is 
_, GCatY -E)O 
ie es ear greta Tema s. , 
Cm, Q- 
m 


7 Cmey Evan 


from which we see that this modulus is not constant but 
changes with the values of Y, the variable. This is merely 
algebraic proof of the irregular nature of the y-scale projection. 
We may observe from the equation for y that it is a fraction 


ZIGZAG AND COMPOSITE NOMOGRAPHS 565 


of the constant distance, Q, between the true origins of the 
outer scales and that y (and hence the scale for the Y variable) 
will always lie between these origins (and hence between the 
two outer scales) so long as the denominator of the fraction 
exceeds the numerator; we likewise observe that the y-scale 
will lie outside of the two parallel scales when the numerator 
exceeds the denominator. 


An interesting thing about the y-scale is its Y-value or. 
calibration at the point midway between the two outer scales, 
that is, when y= 4Q. For this we write 


slg Gp AEA Fae SO Us 
on CRO AG ty EOD Be 
Cm,(Y -E)+Am,=2Cm,(Y -E) 
Am,=Cm,(Y -E) 


Thus at the point midway between the two scales the value of 
Y is always easily found, by dividing the constant coefficient 
of each independent variable by its scale modulus, then divid- 
ing the quotient for x by that for z and adding any added 
constant. If we have the simple case in which both inde- 
pendent scales were plotted on the same moduli, and there is 
no added constant, then the midpoint of the (y) scale expresses 
the ratio of the coefficients in the formula. If the coefficients 
are alike but the moduli are different, and there is no added 
constant then it expresses the ratio of the moduli. Of course 
the addition of a constant, E, merely raises these values by 
its amount. And when coefficients and moduli are alike, and 
no constant is added, the midpoint has the value of 1, and at 
equal distances on either side all the other calibrations will be 
found to be mutually reciprocal. 


Useless as is the mathematical expression for y and m,, a 
similar expression for the temporary projection of the (y) 


566 CHARTS AND GRAPHS 


scale upon the x-axis, by means of which the y-scale can be plot- 
ted without computing, is valuable. We select on the z-scale 
(preferably near its center) a fixed point, », the calibration 
or Z-value of which let us call NV. If we run transversals 
through N and every uniform value of Y to be calibrated on 
the y-scale, we know that the transversals would mark off a 
regular scale upon the x-axis. Let us call the temporary work- 
ing scale w. The intervals or scale modulus, m,, of the w- 
scale, would have the same relation to the modulus of x as 
the calibrations of X have to the calibrations of W. Thus 


Wy, oe ak XxX. 
we ee ahs =" = A 
a eg Now when Z=N, the value of y sas follows 
AX+B =(CN+D) (Y -E) 
CN+D,, CN+D, B 
X= Y - E- 
A A A 


Drop the added constants since they merely shift the zero 
point and 


CN+D 
X= ri ye 
A. CNGED TY 
ote lew Be 
m, CN+D 
M x A 
_CN+D 
w A x 


Or if we wish to work with the length, n, of the plotted point, 
from the true z-axis origin, instead of its calibrated Z-value, 
N, we have 


D m 
=m, (N +=) = —(CN + 
n=m, ( Go? (¢G D) 


If we wish the modulus in terms of the ruler to use, i.e., the 
number of moduli per inch (or per chart-modulus, whatever it 
be), we have 

1 AS ASs 

Ty CHD OOS” 


’ The mathematical steps here are outlined without full details, as the latter would 
make the equations more cumbersome than their importance justifies. 


a 


ZIGZAG AND COMPOSITE NOMOGRAPHS 567 


In short, to prepare a zigzag nomograph for the equation 


i, 
CZ L +D 
we need only compute the following expressions in order to 


plot with an engineer’s hexagonal rule: 


Y Serger . . 
S,=— in which 7, approximates R,=H, —L, 


eae ; : 
S,= * in which r, approximates R,=H, -L, 


in which 7 is the distance (in inches or units of the 
Sy == n chart modulus) of a fixed point on the z-axis from 
2 the origin thereof, 


AS, 


pin which WN is calibrated Z-value on the z-scale. 


The usefulness of the above expressions is in the search 
for ruler-copying values of S which will enable us to plot di- 
rectly from the engineer’s rule. For this purpose the same 
tabular arrangement of columns for the values of t,, 7,, S,, 4S;,, 
Ny Sw, CSS. 7 and t, should be made in order that slightly 
different values of ¢ and r may be tried on various scales. 
If, however, we work with a radiating scale-sheet, then the 
scale-moduli are wanted and these are as follows: 


rye ne 
x R, 
m= 
k, 

Cm, 

iy An. n 
or = CN ey 
A 


The chart will also express the exponential equation 
Abe (cZ+D) A log X+log B 
ix =(EY) OT Sara aac 5 RES 


and amplifications thereof; here one outer scale is logarith- 
mically, the other arithmetically projected. When the other 


=log Y+log E 


568 CHARTS AND GRAPHS 


functions of the variables are used, such as powers, roots, or 
reciprocals, the corresponding projections may be called for. 
In all this work the chart will be seen to handle added con- 
stants with much less trouble than the factorial or logarithmic 


ee 


10 €0—— 80 
9 

ws 16 
8 
? 

cis) 70 
6 
{ a 2 
5 68 65 
4 

60 60 
3 wie 

122 

2 fe 

65 58 
2 

60 50 


Fig. 434. The Y-Scale Outside the Parallel Scales. 

Plotted with S, = —2 (the negative sign shows that scale-values increase upward 
along this axis, instead of downward as normally), Sz = 6, and Sw = .01 for N= 
334. The positive sign of Sw shows that /V-values (on the X-axis) increase (that 
is, become larger positive or smaller negative values) downward, as normally; this 
consequently applies also to the Y-values on the Y-axis. The upward or reversed 
projection of the X-scale (due to the negative sign of Sx) shows that the Y-axis, 
passing through the true zeros (plotting origins) of the parallel axes, lies below 
and outside the two scales, hence the Y-scale is outside the parallel scales. 

This form is not of much value. The dependent variable would better be Y 
when it is used. 


(Note: Above dimensions as of original drawing, here reduced to half-size.) 


parallel nomograph, for it does not require a specially com- 
puted scale for the shifted zeros. The zigzag nomograph is, 
however, on the whole of less value than the parallel nomo- 
graph, for the central axis, being diagonal, is often crossed by 
the isopleths at very small angles, and the readings naturally 
become less accurate. 


ZIGZAG AND COMPOSITE NOMOGRAPHS 569 


Many other forms of nomographs have been devised beside 
the parallel and zigzag forms. The theory and making of these 
involve mathematical work and detail outside the scope of this 
book. They are based upon various geometrical theorems 


65 
60 


66 


60 


Fig. 435. The Y-Scale Inside the Parallel Scales. 


The same equation as in Fig. 434, here plotted with Sy=2, S2=6, and Sy 
= —0.01 for N=334. The positive signs of Sx and Sz show that the normal 
directions of plotting of the X and Z scales obtain, X increasing downward and 
Z upward and the Y axis consequently passing between them. The negative 
sign of Sw shows that the W-values along the X-axis and therefore the Y-values 
along Y-axis, increase upward instead of downward (that is, grow larger positively 
or smaller negatively). 

This form is usually better than that in Fig. 434, for though it gives less 
detail to parts of the Y-scale, it places the dependent variable inside, giving 
more accurate readings. 


(Nore: Reduction of half-size.) 


and are generally built up with straight lines for scales on the 
sides of imaginary triangles and parallelograms. It is indeed 
possible to have nomographs with curved axes but these are 
not often encountered, nor is their need more than exceptional. 
A very large body of the less used nomographs are propor- 
tional, and can be used for equations containing four variables 
which are in or‘can be put into the form of a proportion. 
These charts use two isopleths either parallel, perpendicular, 
or with intersections upon a dummy line, in order to afford 
the readings for the four variables. The interest attaching 
to these less common types of nomographs is still largely 


570 CHARTS AND GRAPHS 


academic; the two simple forms, the parallel and zigzag, afford 
adequate calculating facilities for all practical purposes. 

We have so far considered only the simple forms of these 
nomographs, in which there appear but a single set of three 
axes, and which are suitable only for equations with three 
variables (two unknown). The most interesting form of nomo- 
graph is a compound one composed of two or more inter- 
locking single nomographs and suitable for equations with 
more than two independent variables. Each single nomo- 
graph is a set of three axes, but when two or more sets are 
combined, one of the axes of each st serves double duty, 


E 
1 
nt 
x) 


bog B log SSN 
(218 =) +3 


Pp 


= 


Whi 


oon 
oS é > 
\\ 
t 
= G o 


co 


1 

| 

V/ NN A 
Posi TION /. : 
OF Ws| > 625 5 D _ 
'F L210 ! eta’ ; 

ml N: 

rus VX 

fis 

‘ 


Fig. 436. Construction of Factorial Zigzag Nomograph—Unfinished. 


a 


ZIGZAG AND COMPOSITE NOMOGRAPHS 571 


Nore To Fic. 436 


Showing the working-scale, W, from which the dependent variable scale, Y, is 
plotted; and the true origins of the independent variable scales, shifted for added 
constants. 


The equation is 5a° = (5.71715)¢+2 


in which a varies from 1 to 100, ¢ varies from 3 to 13, and bis to be found. Let 
X =a,Z=c,andY = b, 

Since log 5 + 3 log @ = (c + 2) (log 5.7171 + log 4) 

3loga+ log 5 


or a) = log 6 + log 5.7171 
and the typical formula is 
Hx 
A log X : + log B 
ie = log Y — og £ 
(OV, +D 
Lz 
we have, by substitution 
log 100 
3 log X + log 5 
eet = log Y— (1 log 5.7171) 
zZ | 42 
3 


If we wish the chart to have a total height of T = 5 inches we can piot with the 
following scale moduli: 


Rx = Hx — Lx = log 100 —log1 =2—0=2 Rz = Hz —Lz = 13 —3 = 10 
T 5 iP 5 

— =2.5 inches mz = —— = — = .5 inches 
Rx 2 Rz 10 


mx = 


We must plot the x-scale from a logarithmic scale having one deck for every 244 
inches, and the z-scale from the 20-side of an engineer’s ruler, after allowing for 
the added constant in each case. 

If we select as the fixed point, N, for our working scale, WV, the point calibrated 
as i0 on the z-scale 

CN+D 10+ 2 
My = —————-_ mx = ——— (2.5) = 10 inches 
3 


A 


and we plot the w-scale from an inch-rule (the 10-side of the engineer’s ruler). 
To position the /V-scale we calculate the value of X for any value of Y we choose; 
thus, 


when Z = N = 10\ 1, yee (Z + 2) (log Y + log 5.7171) — log 5 
and Y(= W) = 1f °8* 3 
Pra? Vogl = .75722) — .69897 
‘4 3 
= DIES) 
x =1625.0 


While this value of Y lies outside the range, and is therefore inconvenient, we 
need not recompute Y for another value of Y, but merely extend the 4 scale 
sufficiently to plot /V = 1, after which other /V values follow by the ruler. 


572 CHARTS AND GRAPHS 


being common to two sets and effecting the combination. 
Thus if we have the formula or equation 4=B+C+D+E, 
we will have to break the right side of the equation, having 


Ba> = (6-7171b)(¢ + 2) 


Fig. 437. Construction of Factorial Zigzag Nomograph—Finished. 


The temporary working scale, JV, is erased after the calibration of the y-scale 
therefrom. Also the extensions of scales to the true origins and to x = 625 have 
been erased. Scales have been calibrated on both sides to facilitate readings 
when using an opaque ruler as isopleth. 


four independent variables, into two groups of two each and 
make a parallel additive nomograph of each group, adding a 
third axis to each to express the resultant of each group, and 
then we can combine the resultants in a third parallel nomo- 
graph to show the dependent variable, 4. We would write 


f=B4+C 
and g=D+E 
and A=f+g 


In the first two groups we might let f and g be middle axes, 
but in the third group we would use them as outer axes writing 
A as the middle axis of the group. The order of the axes 
would be B, f, C, A, D, g, E. If for convenience we wished 
A to be the final axis, then we should have to fall back on the 
use of inverted x-scales and carefully arrange the scales so that 


Aor ce 


(0) 
9 
8 
: 7 
6 
5 
4 
=. 9 
2 
1 
Coefficient 
of 
Variables 


’ 


ZIGZAG AND COMPOSITE NOMOGRAPHS , 573 


‘e. 
+ 
a 
. 
* 
a 
7 2 
3 7 Zz 
« 2 reg o e @ Chee s- o 
a5 3 
Swe sod 
e e we es 
° » o 6 e Sc as 
x 2 a = 
= 
8 
S «la ols 
col a aw . 
oo ° °o Ay 
ns) a oOondt a a 2 Ong i) - A 0 a * ” 
sPooar ° : Bee @ a 
an a o 
L3% aw 
° o oem OW o oO 
$a & a ~ x La eo eS at Bre Mee na) 8 op anaes 
ca k a 
aS 
& 
°o 
g 
Lad “owl 
atl ans 
eet Fie 
hae * 
oO |e 
“” “ . 
° 
a of y 
o 
si @ 
ee r 3a3 
~ Om - . . Ae Ecol Bees 
Gee) Sa aed Sar a 0 ek Ya Sa - S> 
mere ee i 
< ° nn 4 oO o wo t m nu a e © 
° : : : : . > es 
A OG 
3a 0 
a su 
e 
° ij 
A a oO o wo + Le) N a ° 
oO Ped 
wo 2 
ee fue 
cao & PPO 
iS © o = ” n - s 


Fig. 438 is a compound nomographic chart by means of which parallel nomo- 
graphs may be constructed. 

Draw isopleths from Rx and Rz on the R scale (first) to Tx and Tz on the T scale 
(fourth) and find scale moduli, mx and mz and rulers Sx and Sz on the Sm scale 
(second). From the latter draw isopleths to 4 and C on the fifth scale and read 
the values of p and q on the third scale. Add the latter, p and g, to get Sy. In 
the above Rx and R:z are the ranges of the two independent variables, x and 2; 
Tx and Tz are the tenths (in inches) to be given these scales on the chart; Sx and 
Sz are the engineer’s rules (or number of units per inch) to use in plotting them, 
A and C are the coefficients of the two variables, x and y; and p and gq are the 
horizontal distances between axes (p between x and y, g between yand z). Sy is 
the engineer’s rule (or number of units per inch) to use in plotting the dependent 
y-scale. Position the scales (for added constants) by a single tnal isopleths. 


Chart to Construct Parallel Nomographs. 


Fig. 438. 


574 CHARTS AND GRAPHS 


8 
ariere es AB, and CB, 
600 
500 
400 
$00 
200 
2 
100 Aes 
DST pa ate See ee 
ORS 60 8 ne ea ee ase ee aro 
50 ce ; 
40 =~ 
30 ae 
= . 
20 . B 
~ =~ 
10 . 
8 ian an $ 
6 ~ 
6 e 
4 oe 
a 
Uf 
$ od 7 
- 8 
2 Be 
9 
10 pee 
i - 
«8 rae rat 
5 o* ian | 
‘5 Ed eocptte --- = 213 mhatoe RA 
4 So ae a 
oe: 16 
a7 
2 18 
2 
pt 
+08 
+06 
+05 
OF 
C t AS, with na Length Csefficients Borking 
Scale omand. tind of Scale Boale 
Rulore om paralled Isopleth in Variables 
frow AS, Inches Point 


Fig. 439. Chart to Construct Zigzag Nomographs. 


Here is a compound nomographiec chart by means of which zigzag nomographs 
may be constructed. Draw isopleths from Rz and Ry on the first scale to Tz and 
Ts on the third scale, and read Sz and Sz on the second scale. From the latter 
draw isopleths to 4 and C on the fourth scale and note intersected points on the 
dummy axis. From the last, the intersection of the dummy scale and the isopleth 
through Ss; and C, draw an isopleth to m on the fifth scale. From the other 
dummy axis intersection, 4 Sz, draw a parallel isopleth to the fifth scale and read 
Sw, In the above 4 and C are the coefficients of the independent variables, x 
and z; Re and Rs are their ranges; Tz and 7; their scale lengths in inches; n is 
the fixed point distance on the z-scale to project the working scale, W, on to the 
y-axis as Y, and Sz, Ss, and Sw are the sides of the engineer’s rule (or number of 


units per inch) to use in plotting. Position the scales for added constants by 
means of these plotting-units. 


ZIGZAG AND COMPOSITE NOMOGRAPHS — 575 


while each y and z scale went in the same direction, each x 
scale went in the opposite one. Compound nomographs can 
be used for equations with many factors instead of terms, in 
precisely the same way, merely using logarithmic projection or 
zigzag nomographic form. The sub-total axes (f and g in the 
example just cited) or the sub-product axes in factorial nomo- 
graphs, are generally left without calibrations, as no one is 
interested in reading their values. They are necessary merely 
as fixation points secured by the first interpolation and fixing 
the isopleth for the next step. They are called dummy axes. 

The fact that the zigzag nomograph performs multiplica- 
tion and addition on arithmetically projected scales makes it 
useful for compound nomographs of formulae involving both 
addition and multiplication. This, indeed, is the chief reason 
for the importance of the zigzag form. Thus an equation of 
the general type 4 = BC +. DE can be solved by the use of two 
zigzags for the two multiplication processes and a parallel 
for the sum of their products. This equation could not be 
shown on parallel nomographs alone, because in them logarith- 
mic projections would have been necessary for the factorial 
processes and the addition of the products, were logarith- 
mically projected would have shown not a sum but a third 
product. 

It has already been said that many other projections can be 
used beside logarithmic and arithmetic ones. Squares, cubes, 
roots, and trigonometric functions can be used. When such 
functions are used, the equations px = mX or x = m,X no 
longer hold, but must be modified to px = mf (X) and x = 
m,f (X). This will require the modification of the calculating 
formulae which have been given for the scale-moduli, but the 
procedure is so similar that it may be left to the devices and 
ingenuity of the chart-maker. Nomograph-making presup- 
poses a fairly thorough understanding of the equation to be 
plotted and with this as a basis, the ingenious experimenter 
will find various and adequate methods of charting. 

Upon the finished chart the scales should be provided with 
titles below or above them, explicitly stating the variables to be 
located or read on each scale. The formula which the chart 
expresses should also be available to the reader somewhere 
about the chart. The best mechanism for the reading of the 
scales is a strip of transparent celluloid with a fine straight 


576 CHARTS AND GRAPHS 

line drawn in ink upon its lower surface. A straight-edge or 
ruler, if possible with a transparent edge, can be used; and in 
an emergency a piece of thread can be drawn tight and held 
for the readings. 


cP 


C7 FITS 


IRL 
17} 


Ef 
i] 


XL 


SS 


1, 


NS 
x 


AS 
SS 


ts 


ZZ 


Plate"B” 


Fig. 440. In Quadratic and Cubic Equations the Position of the Central 
Axes Becomes Variable, and a Chart-field Takes the Place of a Single 
Scale. 


This is the Darville-Johnson Bond-Yield Chart, a chart for determining quickly 
the yields of all types of bonds, including premium bonds, maturing in any num- 
ber of years at practically all coupon rates now in use, including odd fractions. 


—Published by Prentice-Hall, Inc. 


Whole books have been written about the nomograph and 
while it is still a little known chart, yet it is fast increasing 
in popularity, and deservedly so. It is the most easily con- 
structed and accurately read of all calculating charts, and the 
results which can be accomplished with it are always a source 
of amazement to the uninitiated. 


CuaPTerR XLVIII 


SLIDE RULES 


A calculating device which is even more simple to operate 
than the alignment chart or nomograph, but may be considered 
closely related to it, is the slide-rule. We have seen that by 
the use of special projections, the curve of an equation may 
often be straightened, increasing both the ease and accuracy 
of calculations based upon the curve. We have seen that in 
the nomograph the chart field has also been eliminated with 
still further simplicity and benefit. But we come now to the 
slide-rule, in which even the straight-edge (that rudimentary 
substitute for the curve) has been eliminated, the scales losing 
their fixed position with regard to each other and being freely 
movable. The slide-rule is therefore nothing more than two 
movable scales along the same axis, that is, in contact with 
each other. 


We can, however, approach the subject of slide-rules even 
more simply if we consider first the stationary rule. The 
stationary rule is nothing more than a single axis bearing two 
or even more scales, one upon each side of the axis. This is 
the graphic chart of equations containing but two variables, 
(only one independent or known variable). The scales are so 
adjusted that for every value of one variable, calibrated on 
one scale the corresponding value of the other variable may be 
read upon the other scale at precisely the same point along the 
axis. Fixed or stationary scales may be laid off upon logarith- 
mic, arithmetical or any other projections or combinations of 
projections. Such charts are useful in the place of small con- 
version tables, but must be made large or in several segments 
when detailed readings are necessary. [hey can be used as 
ready reckoners for foreign exchange, temperature equivalents 
in Fahrenheit and Centigrade, the conversion of metric and 
common systems of measures and weights, and an infinite 


Salt 


byee et 


CHARTS AND GRAPHS 


variety of similar cases in which the relation between two 
variables is constant and fixed. 


Velooity Force 
40 


Storm 


Gale 


Btrong 
gale 


Moderate 
gale 


Btrong 
breeze 


Moderate 
breese 


Gentle 
breese 


Light 
breese 


Light 
air 


Cala 


THE FORCE OF WINDS 
(Standard Table) 
Fig. 441. 

A Stationary or 
Fixed Rule. 


As its name suggests, the slide-rule is not 
fixed or stationary. If you will take two 
ordinary rulers, one with its scale upon the 
upper edge, and the other with its scale upon 
its lower edge, and bring them together so 
that the two calibrated edges will fit together, 
you will have the simplest form of slide-rule. 
Calculating is done by the simple trick of 
sliding one ruler along the other and reading 
the corresponding values in the new positions. 
Thus in order to add 2 and 4, you need only 
slide the upper rule along the lower one, until 
the zero point on the upper rule is over the 
figure 4 on the lower rule, and then read the 
figure on the lower rule below the figure 2 
on the upper rule. Obviously you have in 
this way added two inches to the four inches 
on the lower rule and you will get six inches 
on the lower rule. The upper rule merely 


~ tells you how much you have added to the 


original distance on the lower rule. Likewise 
to subtract 2 from 6 you need merely place 
the 2 on the upper rule over the 6 on the 
lower rule and read back to the figure 4 on 
the lower rule under the 0 on the upper rule. 
This amounts to deducting 2 inches from 
the original 6 inches on the lower rule, giving 
you a remainder of 4 inches on the lower rule. 
In short, the slide-rule is merely a device 
for the direct addition or subtraction of dis- 
tances, 

In the device just explained, the calibra- 
tion of the two rulers forms an arithmetical 
series and hence the calculating power of 
this device is limited to the processes of ad- 
dition and subtraction. In order to use the 
device for the processes of miultiplication and 
division, of course it is only necessary to 
calibrate the rules upon logarithmic projec- 
tions, so that the addition or subtraction of 


SLIDE RULES 579 


the logarithmic distances will indicate the processes of multi- 
plication and division of the numbers appearing on the scale. 
As in the case of the nomograph, the calibration can be an 
arithmetic or logarithmic projection of sine, tangent, square, 
cube, root, and other functions of numbers as well as of the 
numbers themselves. 

The ordinary commercial slide-rule is nothing more than 
a series of these scales of logarithmic projections of various 
functions, mounted upon bits of wood which fit closely to- 
gether and can be conveniently handled. One rule is made 


© VantdbndiaRinntbtnndeitattldtn taht Lilt tetsPiteletedatetata tt faatusuatuadunbualnfuatintintutit 
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oT Td a an ite rah 
BTA ANTERIOR AAT i ATOR 


SRRTOADY PPO YT A 
Courtesy of Keuffel & Esser, N. Y. 
Fig. 442. A Slide Rule. 
For multiplication of numbers, squares, cubes, tangents, sines and other circular 
functions and also showing logarithms of numbers. 


much smaller and fits within a groove on the other, sliding 
freely back and forth along that groove so that it is not neces- 
sary to hold the two rules constantly together. They are so 
tightly adjusted that the two rules remain without shifting in 
whatever positions they are placed, leaving you free to take 
the readings on the scales with great care. The inner rule is 
called the “‘slide.’” A “runner” is also attached to the outer 
rule for convenience in taking readings, being generally a small 
piece of glass on which a fine hair-line has been drawn at right 
angles to the scale. When you have positioned this runner so 
that the hair-line crosses the point desired upon one scale, 
the hair-line will also cross the desired point upon the 
other scale, and the reading on the second scale can be more 


@ hil 


AIL 


Permission of Keuffel & Esser, N.Y. : 
Fig. 443. The Magnifiers Increase the Accuracy of Readings. 


CHARTS AND GRAPHS 


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SLIDE RULES : 581 


easily taken. And when the two scales which you are using 
are not in immediate contact, but are parallel some distance 
from each other, the runner is necessary to project the desired 
point from the first scale to the second, forming a sort of 
ordinate across the two scales. Magnifying glasses are often 
attached to these runners so as to facilitate more exact readings. 

Because the construction of a straight slide-rule calls for 
rather delicate carpentry,! slide-rules on short notice in the 
home or office are more easily made in circular form. And 
because a circle is endless and over three times as long as its 
diameter, the circular slide-rule can be made on much larger 
scale and with consequently gréater accuracy than a straight 
slide-rule of the same physical size. For the circular slide- 
rule, you need merely pin together through their centers two 


fi 


7 
Rit 
\ 


¥) 
LSS 


zt 
SS 


Courtesy of Keuffel & Esser, N. Y. 
Fig. 445. A Circular Slide Rule—Pocket Size. 


Owing to the great length of a circumference (compared to a diameter), and to 
the overlapping (because endless) edges, the circular rule is very compact. 


circular pieces of paper, the smaller one uppermost, so that a 
scale can be drawn on the visible inner edge of the lower one 
which will always be in contact with a scale drawn on the 
outer edge of the upper disc. Then by rotating one disc above 
the other the readings can be taken off in precisely the same 
way as with a straight slide-rule in which one rule was slid 
along the other. The circular slide-rule operates on precisely 
the same principles as the straight slide-rule, adding or sub- 


1 Excellent examples of straight slide-rules for special purposes may be found in 
the writings of Mr. Carl Barth, The circular slide-rule has been more used by Mr. 


Walter N. Polakoy, 


582 CHARTS AND GRAPHS 


tracting distances around a circumference instead of along a 
straight line, the distances on the scales being prepared by 
angular instead of linear measurement. 


POWER PLANT LOG 
CALCULATOR 
DAY &ZIMMERMANN 
ENGINEERS, PHILADELPHIAPA. 


* 
4 
% 

Z 
fo} 
fe) 
q 
q 

Cc 


73 


3 US 
nN 


hata 
& 


0! 


Fig. 446. A Special Circular Slide-Rule. 
Devised by Mr. Walter N. Polakov—Permission of Mr. Polakov. 


Some difficulty may be met in the calibration of the cir- 
cular scale. Itis comparatively easy to project any scale or 
calibration upon a straight line, but to project it upon a cir- 
cular line or upon an arc of a circle, it is necessary to use an 
instrument for measuring angles, called a protractor. Pro- 
tractors are almost invariably calibrated in degrees, the entire 
circle being divided into 360 degrees. Two other units of 
angular measurement are known, grades (6400 grades to the 
circle) and radians (the radian being the arc equal to the 
radius of the circle), but neither of these is any more useful for 


SLIDERULES* | 583 


the purpose in hand than the degree. Even the metric system 
has no decimal unit of circular measurement. This is unfor- 
tunate because a decimal circular measurement system would 
often be convenient. Sometimes circles are divided into one 
hundred parts for the plotting of 100% circles or pie-charts 
but these are not usually of sufficient accuracy and precision 
to use as protractors. Your best plan in the making of circular 
scales is to decide beforehand approximately how far around 
the circle you wish your scale to run and then turn your scale- 
distances which you would use in calibrating the straight line 
scale into the nearest convenient number of degrees and lay 
them off with a large protractor or scale of degrees, last of all 
re-calibrating the scale for the desired value from your con- 
version table. Thus, if you wish a scale which runs from 0 to 
10 in actual distances to extend about a quarter of the way 
around the circle, you can plot your table of distances directly 
onto the circle from your protractor by using that portion of 
the protractor which extends from O degrees to 100 degrees, 
but if you wish your scale to extend over half way around the 
circle you must first double the actual distance values before 
plotting them as degrees, so that you can plot through the 
protractor from O degrees to 200 degrees. 

Circular slide-rules can be made with a number of inde- 
pendent scales, each sliding on separate pieces of paper but 
all pivotted together at their centres by a small rivet. A sub- 
stitute for the runner can be attached in the form of a strip 
of transparent celluloid with a fine ink line drawn radially 
from the center or pivotal point. This ray can then be swung 
about the circle and laid over any desired point on the scale 
to facilitate readings in the same way as the runners on a 
straight slide-rule. Another device is to make the uppermost 
circular sheet of paper so large as to cover all the other sheets 
and then cut windows or circular slits in the upper sheet where 
the lower scale should be seen and mark small pointers or 
arrowheads next to the windows for readings on the lower 
scale. 

Circular slide-rules are easily made when you have once 
grasped the fundamentals of their construction and they afford 
the greatest play for ingenuity. When skilfully made, they 
perform the most intricate calculations with astonishing ease 
and simplicity. Their only limitation appears to be that they 
perform in one operation only that particular type of mathe- 


584 CHARTS AND GRAPHS 


matical process for which they have been designed. As they 
operate by adding and subtracting distances, they will perform 


EDITION 


S¥vr0d GAS 


tee Ie 


Fig. 447. A Circular Slide Rule with Many Variables. 
Showing Cost of Book Printing, 


addition and subtraction if the calibration is in terms of 
arithmetical series, that is, in the natural numbers. They will 
perform multiplication and division of as many factors as 
there aze scales, by the simple trick of calibrating them log- 
arithmicaily, that is for the logarithms of the natural numbers. 
But they cannot be used at the same time for addition and 
multiplication, for the two processes require different types of 
calibration or projection of scales. This limitation is ordin- 
arily not a serious one, because most tedious business compu- 


SLIDE RULES 5804 


_ tations are either processes of addition or multiplication but 


not combinations of the two.2 


(Se GS Ro. 
s) 
were Boa, 


BuOEry 
Y4vg 


Fig. 448. The Same as the Preceding, Except that All Scales are 
Covered and Seen Only Through Small Open Slots or Windows. 


When facilities are at hand for delicate and precise car- 
pentry work, the straight slide-rule principle can be elaborated 
and developed by a series of pulley wheels with cords passing 
over them and connecting movable pointers along separate 
fixed scales. The pointers can then be adjusted for the par- 


2 The scale-moduli of slide-rules vary inversely with the coefficients (in additive 
rules) and exponents (in factorial, or log, rules) of the variables, and are alike as 
these are alike. The length of the rules or scales, therefore, varies directly with the 
ranges of variation of the variables (unless the moduli are unlike). 

In the pulley-wheel and pointer type of slide-rule next described, the pulleys can 
be made with various leverages and thus modify the moduli, affording opportunity 
for the adaptation of the lengths of scales to any moduli, range, or length, desired. 


586 X 


5 1 2 S \0 2 §o {00 "tooo 10,0v0 100,080 


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iy 
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12 ! 7 
1 
I le 
il | I 
I 1 teas 


LLLLLLZTZ LLL 


LZ 
U to: ‘ 


Fig. 449. An Arrangement of Pulleys, Wheels, and Weights, by Means 
of which the Pointers come to Rest at the Roots of the Equation. 
(See note on opposite page.) 


SLIDE RULES 587 


ticular readings of the variables and the final pointer will come 
to rest at the correct reading of the answer for the equation, 
the product or quotient of the various component factors. In 
such devices it is even possible to combine the different proc- 
esses of addition and multiplication in one machine at the 
same operation, so that machines can be constructed to give 
the answer for any set of variables in any equation. It is 
hardly in the province of this book to go into the details of 
construction of such calculating devices. The circular slide- 
rules are, however, so easily constructed and sometimes such 
great labor-savers and time-savers that it is well to be able to 
make them, when occasion arises, especially adapted to your 
own problem.3 . 


3 It is by this time doubtless apparent that slide-rules and nomographs are clearly 
akin. When we have an equation with one independent variable, we have a fixed and 
rigid equality between it and the dependent variable, by which one is always a certain 
function of the other. In such cases, the chart of the equation is a chart with fixed 
or stationary scales. But when there are two independent variables, we can either 
(1) use sliding-scales, so that one of the variables can be eliminated by a proper setting, 
or (2) use separated scales (that is, nomographs) so that one end of the isopleth can 
be properly set to eliminate a variable. And over against these uses of scales, we have 
always a more cumbersome alternative, either for one or two independent variables, 
in the curve-chart. 


Note To Fic. 449 
All wheels are fixed except the three in frames, which slide up and down. The 
shaded bars are weights free to slide up or down, and sufficient to balance each 
other, so that the string remains taut and at rest wherever it 18 set. The small 
cross in the lower left-hand corner is the only fixed point to which any string is 
attached. Computing is done by setting any pointers at proper points and 
reading the answer on the remaining pointer, 


CHAPTER XLIX 
HUNDRED-PER-CENT TRIANGLES 


Whenever you are dealing with problems in which three 
elements or parts combine to form one whole, and you are 
interested not in the whole but in the proportions of the three 
parts, the computing of the problems as well as the recording 
and presentation of the results can be accomplished with a 
chart which we may call the 100% triangle. The chart is 
also known as the trilinear chart and its rulings are sometimes 
called areal co-ordinates. It differs from the 100% bar in 
that it is limited to divisions of a total into three and only 


xfytsca 
Fig. 450. 


three components, but it is similar to the 100% bar in that it 
does not distinguish between large and small totals, all totals 
588 


HUNDRED-PER-CENT TRIANGLES 589 


being reduced to 100% and appearing upon the chart in exactly 
the same size. It resembles the nomographic charts in that 
it is a real labor-saver in the work of computing. 


Like the nomograph, the 100% triangle is based upon a 
trigonometric principle. The theorem in this chart is that in 
an equilateral triangle the sum of the three perpendiculars 
dropped from a point within the triangle to the sides of the 
triangle is constant and always equal to the altitude of the 
triangle. The rule is limited to equilateral triangles, and the 
100% triangle is therefore always made of equilaterals.1 The 
three perpendiculars bisecting each side and extending from 
the sides to the opposite angles of the triangle form the three 
“axes” of the chart. As in the calculating charts, the word 
“axis” is here used in the special sense of a straight line to 
which a scale is attached and from which ordinates are pro- 
jected normally (that is, in perpendicular direction). Rulings 
parallel to the sides serve to project the scales across the chart 
in the manner of co-ordinates. A little thought will show 
that the 100% triangle is merely a variety of the rectilinear 
computing chart, in which the zscale is calibrated so that 
z=100 —(«+y) instead of z=x+y, or that 100=x+y-+z. 
From this it follows that the equilateral triangle is not essential; 
any triangle can be used, but for reasons already pointed out? 
the scales are most easily projected upon the equilateral form, 
since then the distances along the three axes have the same 
real significance. 

The scales used for the axes of the chart may have any 
range, but the customary scale is an arithmetical one, cali- 
brated in percentages and ranging from zero to one hundred 
per cent. This forms the arithmetical 100% triangle. Data 
to be charted upon it must first be turned into percentages of 
the total of the three elements charted. The chart is an addi- 
tive one, the three elements combining by addition to form a 
total. As has been said, the chart does not distinguish between 
the sizes of the totals, but shows them all of the same size. 
Nor is it possible to attempt to show relative total sizes by 
varying the dimensions of the equilateral triangle, as its area 


1 Obviously, any other triangular form can be used, but the intersections of the 
three sets of co-ordinates become less sharp and well-defined, and the scale-moduli 
become less easily commensurable. See Chapter XLIII. ; 

2See Chapter XLIII. The 100%-triangle is most fully described in Haskell, 
Allan C., How to Make and Use Graphic Charts, Codex Book Co., New York. 


590 CHARTS AND GRAPHS 


increases by the square of the increase in its altitude. The 
general form of the equation for the chart 1s 100% =x+y +2”. 


¢ 
% 


Fig. 451. 


The classical example of the use of the 100% triangle is 
the analysis of food values in terms of calories (or heat pro- 
ducing units) of fats, proteins, and carbohydrates or hydro- 
carbons (sugars and starches). The well-balanced ration 
being about 20% proteids, 60% hydrocarbons and 20% fat, a 
point can be located on a 100% triangle at the intersection of 
these three ordinates and the approach of various foods and 
combinations of foods to this ideal easily seen. Moreover, 
the computing to plan a well-balanced meal can easily be done 
upon the chart. Thus with the two points for bread and milk 
plotted upon a chart, a line drawn between the two points 
indicates all the possible food values which can be had by 
mixing the two in various proportions. With equal caloric 
amounts of each, the point midway upon this connecting line 
will give the components of the combination. Foods known 
to lie upon the opposite side of the ideal ration-point from 
this mixture-point, must then be added to bring the meal 
nearer to the ideal, a straight connecting line again serving 
to show the results of combinations in all possible proportions. 
Such calculations can be performed upon this chart in a frac- 
tion of the time which they would require by any other method. 


HUNDRED-PER-CENT TRIANGLES sgl 


In business this paper can be used in innumerable cases 
vhere a total is divided into three parts. Advertizing appro- 


Fig. 452. The Hundred-Percent Triangle for Food Values. 


The chart shows the fats, proteids, and carbohydrates (in calories) of chicken 
bread and milk, the full lines show the results of mixing any two of these, and the 
intersection of the broken lines shows the result of combinging an equal quantity 
of each.— Permission of Mr. Malcolm C. Rorty. 


priations, for example, may be divided into magazine, news- 
paper, and outdoor advertising. Salesmen are often expected 
to preserve a certain proportion between their sales of large, 
medium, and small profit lines. Inventories may be kept in 
terms of raw materials, finished stock, and goods in process. 
Assets are often divided into current, fixed, and intangible 
assets; costs into payroll, materials, and overhead. In scien- 
tific work the chart has been used for the chemical analysis 
of mixtures of three elements. Extensive use of the chart has 
been made in engineering, for comparing various grades of 
coal as to their hydrogen, oxygen, and carbon content; and 
for investigating concrete mixtures and so on. In economics 


\ 


592 CHARTS AND GRAPHS 


the chart would seem admirably adapted to the study of 
projects for the joint representation of owners, workers, and 


Compromised 
0. 


WAV 

BVA OAS GAN 
OS. 
? s 
AINA eX 


a D 


t°) 10 20 30 40 50 60 70 
SETTLEMENT OF STRIKES 

Percentages of Victories for Employers and Employees 

a ikes 


nd of Compromised Strik 
United States 


1916-1921 
(Arranged from Monthly Labor Review) 
Fig. 453. 


public in industrial disputes, or the division of earnings into 
wages, salaries and dividends, or the division of authority 
between management, workers, and stock-holders. ‘The tri- 
partite division is frequently called for. 

By using logarithmic scales, the chart can be made fac- 
torial instead of additive and used for cases where three 
elements combine by multiplication to form a product. Its 
equation is: log 1=log «+log y+ log xz, or 100%=-xyz. 
As before, the chart will not show the size of the resulting 
product, it will merely show the relative proportions of the 
three components. The scale can be of as many logarithmic 
decks as desired, according to the range of variation of the 
components, but of course the same scale must be used for all 
three axes. This chart is often better when its scales are re- 


HUN. DRED-PER-CENT TRIANGLES 593 


_ calibrated to absolute quantities, the product of which is a 
fixed given amount. The chart can then be used for the com- 


DANA Oe 
CT RRO 
HE x7 


EE NOES 


XYZ = 100 
Fig. 454. The Factorial 100% Triangle. 


parison of the component factors in two or more equally large 
products. Its equation then is: log C=log x+log y+log z, 
or C=xyz. 

The logarithmic 100% triangle has apparently never been 
used, but it is almost as often desirable as the arithmetic one. 
In engineering, for example, there are innumerable equations 
involving three factorial variables. A most obvious case is 
that of cubic measurements involving height, length, and 
breadth. The commercial measure of electric current is the 
kilowatt-hour, a product of voltage, amperage, and hours. In 
factory management, labor cost for a job is the resultant of 
the number of workmen, their average hourly wage, and their 
time on the job. In finance, business, and economics, similar 


594 CHARTS AND GRAPHS 


(x) 
\AAXRE ERED 


1 +2 91.0 


X and Z 
XN¥ 2 1,0 
Zz 


Fig. 455. A Single Scale Used for Two Axes. 


ov MOK WAN 
=e Wy) NN 
OETA 


Length and Breadth 


DIMENSIONS OF BEAMS OP EQUAL TRANSVERSE STRENGTH 
(Beams of rectangular section) 
Breadth x (Depth)* 23 
eet 
Length 


Fig. 456. 


HUNDRED-PER-CENT TRIANGLES 595 


equations will be met, involving three variable factors, and sus- 
ceptible to analysis by this form of chart. 

With a thorough understanding of the method of the chart, 
it can also be used for division, by the use of reciprocals. The 


equation then is log C=log x+log y —log z, or c=". An 


equation of the form C == or C “> could equally well be 


shown. Without logarithms, the chart thus performs sub- 
traction, as 100%=X+Y -—Z or 100%=X -—Y -Z or 100% 
= —Y —~Y-Z,. There is also another method for these cases, 
which does not involve recalibration of scales. It is based 
upon the more general geometric theorem that the algebraic 
sum of the perpendiculars from the sides of an equilateral 
triangle (or extension of the sides) to any point in the plane of 


P 


Q:(-x)ry + 2) Ps (x) + y+2 = 100% 
=100% 


Fig. 457. 


the triangle, is constant and equal to the altitude of the triangle 
The method involves the use of the area outside of one of the 
sides of the triangle, and hence. on the axis of that side, in the 
negative part of the scale. 


596 CHARTS AND GRAPHS 


In common with the 100% bar, and the 100% circle or 
pie-chart, the additive 100% triangle is equally significant as 


VAVAVAVAVAVAVAVA™A\ 
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WAVAS VV, 


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X-Y422=100% 
Fig. 458. 
to the linear measurement and the area measurement of i its 


parts. For the plotted point within the triangle can be con- 
nected with the three angles by three straight lines, and these 


cue 
XK 
EAVES 


SERLES 


PRRBOORA 


NZ in Process 


Fig. 459. 


straight lines will be seen to break up the triangle into three 
small triangles, whose areas are in the same proportion as 


HUNDRED-PER-CENT TRIANGLES 597 


their altitudes, or distances along the scales of the triangle. 
In this case, as in the 100% bar and pie-chart, the area of each 
segment is significant because only one of its dimensions varies, 
the other dimension being constant for the three segments or 
areas. Obviously, the segmentation of the 100% triangle is 
only useful in additive charts and only desirable for extremely 
popular use, when the chart is to show but one set of data, 
that is, one sum broken into three parts. 

For both additive and factorial equations, with either 
absolute or relative values or units of measurement, the chart 
will be found strikingly illuminating and an ssailein means 
of analysis and comparison within its somewhat narrow 
limitations. For it must always be remembered that the chart 
does not show totals, products, or other resultants. It shows 
only the comparative sizes of the components. If, therefore, 
it is the size of the resultant in which you are interested, the 
chart is worthless, but if it is the proportions of its three parts 
or factors, the chart is admirable, and is in fact, quite the 
clearest possible means of analysis, 


Sifertan\, acta ee 
aeos, ALES on { 


Sitter @) 


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a 


PART VI. TWO- AND, THREE-DIMENSION DATA 


CuHapTer L 
HUNDRED-PER-CENT SQUARES 


We have so consistently inveighed against the use of areas 
to illustrate quantities that the reader will indeed be surprised 
at some coming retractions, however guarded and limited 
we may make them. But the fact is that we now propose to 
turn to advantage the very feature of areas which has previ- 
ously been their greatest fault. Let us examine this feature 
closely and see how it can be done. The reader is, of course, 
familiar with that elementary theorem of geometry and arith- 
metic, which states that the number of square units of meas- 
urement in an area is a product of the linear units of measure- 
ment in its two dimensions. He therefore realizes that the 
variation of both linear dimensions by a given ratio results 
in the variation of the area itself by the square of this ratio. 
This has been the dangerous stumbling block in the way of 
using areas in charts which are intended to illustrate but a 
single ratio or set of ratios, To prevent the confusion and 
doubt which might arise in the minds of those who see our 
charts, it has been necessary to maintain between the areas 
the same ratios as exist between their linear measurements. 
And to maintain this identity of ratios, we have been obliged 
to keep one linear dimension constant wherever ‘areas have 
appeared. In the 100% bar, the 100% circle, the 100% tri- 
angle, and in bar-charts, and even band charts, in short in 
all charts using areas up to this point, we have invariably 
striven to maintain one constant dimension, with this specific 
purpose of making variations in areas equal to variations in 
the one varying dimension. 

We now come to data in which we wish to show simulta- 
neously three ratios or sets of ratios, one of which is always 
the product of the other two. In other words, we wish to 
show two factors or sets of factors and their product. And 
to this purpose it is obvious that the area is excellently adapted, 


598 


HUNDRED-PER-CENT SQUARES 599 


by reason of the feature which has just been described. What 
was previously an obstacle to the use of areas (varying along 
both dimensions) now becomes an essential advantage. And 
as in the case of bars (that is, areas varying in one dimension 
only) so also in the case of areas proper (that is, areas varying 
along both dimensions), we find that the charts can be di- 
vided into two groups. The first group is composed of charts 
in which the total or whole area is a constant and is cut into 
segments whose sizes are of interest to us. The second group 
is composed of a series of separate areas, which may or may 
not be individually segmented, but in which the total sizes 
of the individual areas vary. In the section on bars, the first 
type was called the 100% bar; the second, a bar-chart. Simi- 
larly in this section the first type is called the 100% square 
or rectangle; the second, an area-chart or area-bar-chart. 

Before proceeding to the separate consideration of these 
two types, we must call attention to a general limitation hold- 
ing for all types of area-charts. These charts, as we have said, 
illustrate simultaneously two factors and their product, the 
product being shown by the area itself, the factors by linear 
dimensions. Now it has already been frequently pointed out 
that the human eye cannot so easily or precisely judge of 
square measures as of linear ones. Hence we must expect the 
illustration of the factors to be clearer and more easily evalu- 
ated by the reader than the illustration of the product. And 
we may say in general, therefore, that the area charts are 
desirable only for data in which thé product itself is of less 
importance than one or both of the factors. Where an ex- 
plicit illustration of the products is necessary, affording pre- 
cise and detailed comparisons of the products, this chart does 
not suffice, but where the primary importance attaches to 
one or both of the factors, and the product is only of sec- 
ondary importance, the chart will serve excellently. 

Like the 100% bar, circle, or triangle, the 100% square 
is a device for the illustration of the parts of a total. Unlike 
them, however, the 100% square does not show only one 
classification of the parts, but shows simultaneously two inde- 
pendent classifications, which combine factorially to produce 
a great many small parts. The data for the 100% square, 
therefore, consists of two interlocking or mutually crossing and 
subdividing classifications of the parts of a whole. Whatever 
the absolute value of this whole may be, its relative value is 


600 CHARTS AND GRAPHS 


100%, and the chart is ordinarily calibrated in percentages. 
If the data is not already in percentages, it can easily be 
turned into percentages to facilitate charting. The important 


| Proprietors |cierks and | Skilled Semi- Leborers 
Managers kindred Workers skilled and 
Officials Workers Workers Servants 

41,609,192 | 11,165,536 | 5,638,144 | 4,914,651,| 6,384,567 | 15,506,206 | 


5,449,332 

553,203 
3,215,721 
1,530,704 


Total 


Total 


Agriculture, forestry and pueeeret 


animal husbandry 
Extraction of minerals 1,090,854 
Manufacturing and mechanicel 12,812,701 
industries 
Transportation 3,066,305 


§,501,742 


50,041 
4,247,232 


28,610 
660,622 
206,352 


4,689,126 
225,525 


455,505 648,119 


Trade 4,244,354] 1,615,823 | 2,062,864 355,205 210,442 
Public service (not 7 20 658,351 112,769 
elsewhere classified) Mer " , 

Professional service 2,152,464 2,128,787 ce 


365,249 | 2,434,123 


3,119,955 | <6 


3,119,955 


Domestic and personal service 3,400,565 
Clerical occupations 


OCCUPATIONS OF THE GAINFULLY EMPLOYED POPULATION 
(10 years of age and over) 
The United States 


19 
(Source:- U. S. Bureau of Lat’ r Statistics) 


Fig. 460. The Original Data for a 100% Square. 


thing about the data is that it should be clearly arranged in 
a tabulation or table, with the items of one classification listed 
down an edge of the table as the stubs of the table, and the 


I 5 
Total Proprietors 


Clerks and| Skilled | Semi- | Laborers | 
Managers kindred | Workers | skilled and | 
Officials Workers Workers | Servants 
Agriculture, forestry and 26.31 ints 
enimal husbandry . 
Extraction of minerals 2.62) 
Manufacturing and mechanical 
industries Bees 
Transportation 7.37 
Trade 10.20) 
Public service (not 1.85 
elsewhere classified) ad 
Professional service 5,18 
Domestic and personal service Baad 


Clericel occupations 7.49) ‘ ese | 


OCCUPATIONS OF THE GAINFULLY EMPLOYED POPULATION 
(10 years of age and over) 
The United States 
1920 
(Source:+ U,. S, Bureau of Labor Statisties} 
(percentages) 


Fig. 461. In this Form the Data is not Chartable. 


items of the other classification listed across the top of the 
table as its column headings. In the body of the table, at the 


os 


A 


HUNDRED-PER-CENT SQUARES 601 


intersections of columns and rows, are placed the detailed 
figures which correspond to both classifications. , 
In turning the absolute detail figures into percentages of 
the total, we have, of course, no trouble, merely dividing each 
figure by the figure for the total. But in this form, the data 
is no longer factorial, the detailed figures not being products 
of factors which are known, and therefore not being amenable 
to charting by areas. To draw areas we must know the factors 
which will be plotted as the linear measures along the two 
dimensions of the area. It is therefore of no use to us to turn 
the absolute values into direct percentages of the grand total. 
Instead, we turn the sub-totals for each row or column into 
percentages of the grand total, and then turn the detail 


| Total Proprietors |Clerks and|Skilled | Semi- | Laborers 
Managers kindred |Workers | skilled and 
Vertically | Across|| Officials Workers Workers |Servants | 
| 100.0 100.0 26.84 13.54 11.83 15.33 32.46 | 


100.0 50.35 --- aoe 


Total 

Agriculture, forestry and 
animal husbandry 

Extraction of minerals 


Manufacturing and mechanical 
industries 


Transportation 


Trade 

Public service (not 
elsewhere classified) 

Professional service 


Clerical occupations 


Domestic and personal = 


e 
OCCUPATIONS OF THE GAINFULLY EMPLOYED POPULATION 
(10 years of age and over 
The United States 


1920 
(Source:- U, S. Bureau of Labor Statistics) 
(percentages) 


Fig. 462. Here Each Row Totals 100%. 


figures in the body of the table into percentages of the sub- 
totals for the rows or columns in which the detail figures 
occur. Thus we make the detail figures in themselves products, 
that is, percentages of percentages. 

In this step we come to a choice between turning the de- 
tailed figures into percentages of the sub-totals for the columns 
or into percentages of the sub-totals for the rows, in which 
they occur. One or the other must be used, that is, either 
the sum of the detail percentages in each column must add 
up to 100%, or the sum of the detail percentages in each row 
must add up to 100%. We cannot expect that addition up 
and down by columns, and addition across by rows, will both 


602 CHARTS AND GRAPHS 


give 100% in the same table. We must, therefore, make a 
distinction between the primary classification, in which the 
sub-totals are percentages of the grand total, and the sec- 


lProprietors|Clerks and] Skilled| Sem1- 
| Managers kindred | Workers| skilled 
Workers 


Laborers. 
and 
Servants 


(across 
Total (vertically 
Agriculture, forestry and 
animal husbandry 
Extraction of minerals 


Mamifacturtng and mechanical 
industries 


Transportation 


a 


Trade 


Public service (not 
elsewhere classified) 


Professional service 


Domestic end personal service 


Clerical occupations 


OCCUPATIONS OF THE GAINFULLY EMPLOYED POPULATION 
(10 years of age and over) 
The United States 
1920 
(Source:- U. S. Bureau of Labor Statistics) 
(percentages) 


Fig. 463.- Here Each Column Totals 100%. 


ondary classification, in which the détail figures are percent- 
ages of the sub-totals. It is not a matter of importance how 
we place these classifications, but as a general rule in tables, 
the primary classification should be listed in the column- 
headings and the secondary classification in the stubs, to 
facilitate checking up on the computing. Where one classifi- 
cation is much more lengthy than the other, it is of course 
generally more convemient to arrange the longer classification 
in the stubs and the shorter one in the column-headings. In 
charting, the rule is generally reversed, the primary classifi- 
cation being shown along the vertical axis of the chart and 
the secondary one being shown horizontally. The chart itself 
always shows very clearly which classification has been made 
of primary importance and which of secondary importance. 
It often happens that both classifications appear on their 
merits to be equally important, but it is nevertheless. necessary 
that the distinction be made and the data must be prepared 
in the form described before the chart can be made. 

The chart is made by laying out a square with co-ordinate 
rulings. Along both axes of the square, that is, along its vertical 


HUNDRED-PER-CENT SQUARES 603 


and horizontal edges, a scale is marked off in percentages from 
0% to 100%. Arithmetic projection of the scales is used only 
in area charts; and in the usual form, that is, in the truly 
square-shaped chart, both scales are identical. The primary 
classification of the grand total is laid off upon the vertical 
scale by means of horizontal lines extended across the chart 
to form layers of a uniform length but of varying widths or 
depths. 

At this stage, the chart reminds us of a 100% bar turned on 
end and made very short and thick, for the chart bears as yet 


Agriculture, forestry, ai rei 
@ninel husbandry ser’ ayes ond 


Butrection of ainerels Extrestion of ainerals 


Werafecturing and mechanical Mamfeeturing and aechentea) 
Induotrios dodustries 


‘franaportation ‘Transportetion 


Trade ‘Trade 


Public service (not Fublie -ssrvice (not 
eluewnere classified) elsewhore elesgified) 
Professional service Professional service 


Domestie and personal service Domeptic ond personal service 
Clerical eccupetions e Clertee) cosunstions 


OCCUPATIONS OF TUE CATNFULLY EMPLOYED POPULATION 
(10 yeurs of age ) 
The United States 
2 


(Source:- 0. S. Bureau of Labor Statietics) 


Fig. 464. The Primary Division Alone Plotted from Fig. 462. 


but one classification and its segments or layers show, both 
by their depth and their areas, the figures for this one classifi- 
cation of the parts of the grand total. The layers, however, 
need not be distinguished by colors or shadings, for as will 
shortly be seen, they will be sufficiently distinguished by the 
markings of the secondary classification. The next step is to 
enter these secondary classifications. Each layer is now 
treated as a separate 100% bar and divided up as indicated 
by the detail figures in the body of the table of data. Notice 
that each layer is separately segmented, by vertical dividing 
lines which may or may not vary in their positions from layer 


604 CHARTS AND GRAPHS 


to layer. These segments of the layers are now colored or 
shaded to distinguish then, and the chart is complete. A key 


ESE NI AS 
A IRV Ae If W) NZ 


A, 


— ( (at oy 
‘ try, one FT fs MSE SSA fertculture, forestry, end 
ray = ge a reeaats oa heceee 
ies 


EIS 
TS (Zz 


TG; 


Extraction of mineralo [YP Brtrection of minerals 


WILE 
FA %e1E 


Memufacturing and meshaniead 


suite Mamfecturing ond rechanicss 
industries dorkwre industries 


TTT TTT Tro 
Tr 


Trensportation | Transportation 


4 
ZA 
Trade bj trade 


Podlic service 
(not elsewhere classified) 
Frofespionel cervice 


Public service 
(not elaswhers olesettied) 
Professions) service 


Daavatic and personal service Domeatio and personel servdee 


Yj 


Clerical cceupetions 


INN 
BEF RRR IES ERTS ER 
‘WM 


OCCUPATICNS OF THE OALNFULLY EMPLOYED POPULATICH 
(10 yooro over) 
Tn 


Clerics) sooupstions 


we United Stetew 
1920 
(Sourqgi= Us S. Bureeu of Labor Statistica) 


Laborers 
Managore ekilled 
ofticlele Sorkere % Servants 


Fig. 465. The Completed Square. 


to the shadings should be added, to guide the reader of the 
chart. And it will be seen that the area of each shaded seg- 
ment of a layer is to the area of the entire square, as the ab- 
solute value of each detail figure in the body of the table of 
original data is to the absolute value of the grand total. 
Various modifications of the 100% square are sometimes 
useful. The scale of the primary classification may be cali- 
brated in absolute values instead of percentages. In this case, 
the square-shaped outline of the entire chart is often discarded, 
and the chart made rectangular, thus becoming a 100% 
rectangle. This is often done where the detail of secondary 
sub-divisions is great and the areas of segments would be so 
small as to be ill shown except by enlarging one of the scales. 
Either scale may be enlarged in this way, according to the 
nature of the data. It is also possible to project the primary 
classification upon the horizontal axis and the secondary one 
vertically. In this form, the chart strongly resembles the 


UVYIL) Domestic and personal servic: 


HUNDRED-PER-CENT SQUARES 605 


staircase relative band curve chart with ordinates at irregular 
intervals. The rectilinear lines used to segment the layers 


Proprietors Clerks dnd Skillad Semt- Laborers 
jenagers kindred Workers 
ortietaln rkers 


1 : 
I derteullure, forestry, Ae 
(7) Agriculture, beat and Agrtoulturs, forestry, and animal busbeodry 
an 


me} auebandry and apinal husbandry 


riculture, forestry, amt (7) 
aninal husbandry ti 


Extraction of ainarale (If? 


(11) Extraetion of sinerals p———— 


(IIT) Mamufacturing and mechanical 
industrics 


Wanufactor tng and meohenion! 
(1¥) Transportation areas Minny 


(V) Trade 


(vn) pueise nervics ( Transportation (IV) 


now 
sowhere classified) 


Trade (¥) 
Public gorvice (now 

Slaavhore olessitiaa( (¥I) 
Profsssional service (VII) 


(VIL) Professional service 


Domestic and personal serviee( IX) 


Servants 


Proprietora Clerks and Skilled Semi- Ty E 
(IX) Clerical sccupetions Wer akere Victron | a "eorkers ra tan panorers Clerioal occupations (Ix) 
ofticiels Workere forkers: Servants 


° 


OCCUPATIONS OP THE GAINFULLY 
(10 years of age 
? s 


(Source:= 0, S. Bureau 'of Labor Statistics) 


Fig. 466. Here the Primary Division is the Horizontal One, 
Plotted from Fig. 463. 


form staircase curves separating the colored or shaded bands 
which form the sub-totals according: to the secondary classi- 
fication. 

The 100% square or rectangle becomes identical with the 
relative band-chart (with staircased curved) when the primary 
classification has a numerical basis, forms an ordered math- 
ematical series; and can be called a variable. So that the 
whole subject of relative band curves charts may indeed be 
considered a detail of the 100% area. Frequency series fall 
particularly well under either head. And because the pri- 
mary classification can be shown upon either axis of the 100% 
area, it often happens that what is really only a relative band 
frequency curve chart turned upon edge, seems at first to be 
a new kind of chart. When smoothed curves are substituted 
for the staircase curves, by connecting the mid-points of the 
secondary segmenting lines, of the 100% area, the highly de- 
scriptive name of ‘“‘marble-cake chart” has sometimes been 
used for the resulting picture. This is a very interesting form 


CHARTS AND: GRAPHS 


606 


he area- 


into t 
representation in this form, but when the change in the sec- 


ly slight errors creep 


10US 


a 


of the 100% area. Obv 


“op8uepey %00L V “L9b “314 


(TY Jo uoTEstumoD TeTuysnpUl -:e0uN0g) 
6T6T 
oTuO 
SHINLSAGNI ONIYGLOVEONVA NI SHOVR 


(0384 Youg woTeq eequecseg) 
0OT 06 03 OL 09 os OF oc oz ot oe 


Bisutee-eFey BLoureC-eFEy 


SSSLW’™LUL_QQ Ap) 


BYLETO edTFIIO BAL9TO EdFJJO 


pus ‘sucudeazoueys ‘sxedeamjo0g a # = INS ‘ LBZ B puy ‘sueydeuSoucyzs ‘sredeeqx00g 


(Buz T{saery you) etdoed setes ea Reread el U2777 777A aie (FUT TTesesy you) eTdoed setes 
zen ost Off seg : Tes erg ote ef og 


BAUTTOg JO eTeos 


ondary classification is real] 


y gradual and not abrupt, the 


chart has gained in interpretative powers. 


© 


It is commonly 


HUNDRED-PER-CENT SQUARES 607 


useful in the analysis of the component parts of a frequency 
series, the plotting of the independent variable (or primary 
classification) along the y-axis affording greater popular appeal 
through the coincidence of increasing numerical values and 
rising position upon the chart. 

The 100% square can be used for data which is not classi- 
fied by, or dependent upon an ordered numerical series and in 
which there really are no two interlocking schemes of classifica- 
tion, but merely one independent variable. In such cases scales 
are useless on the chart and the chart itself is wholly pictorial. 


Other 


War Work 


At tho front 
1.3% 


6.8% 


Women in 


productive work 


Ab R 


Men in 


productive work 


17.8% 


Old men and boys 


22.9 % 


Fig. 468. A 100% Square. 
Showing Wartime occupations of the population, U. S., 1918, according to official 
estimates; taken from the Annual Report of the Secretary of War for 1919. Total 


population, 105,000,000. 


It has already been said that area charts must be projected 
arithmetically upon both axes, for the reason that only upon 


608 CHARTS AND GRAPHS 


this projection does the area itself illustrate the product of 
the linear dimensions. When a ‘‘marble-cake chart,” for 
example, is drawn with its primary classification upon a log- 
arithmically projected scale, the areas upon the chart lose 
their significance, and the chart itself really becomes merely 
a chart of frequency curves. The logarithmic projection may 
be necessary to straighten the curves or to show parts of the 
data in sufficient detail, but great care must be exercised that: 
the reader of the chart should not, under these circumstances, 
attach the slightest importance or significance to areas. 

The student who has noted how the 100% bar is particu- 
larly adapted to showing the division of a whole into two 


UNITED STATES 


47 Te 


CANADA 


18% 


GERMANY fran! GR. BRITAIN 
: 6% Cor 
[smrrzaeneed 


OTHER UNION OF SOUTH 
counties] COLOMBIA AFRICA 1% 


SIBERIA 
22% 


WNOO- CHINA 
3 


AUSTRALIA 
2'4, To 


| 


Fig. 460. 
Showing the estimated unmined coal supplies in 1920. 


parts (though it can show any number of parts) and how the 
100% triangle is particularly adapted to showing the division 


HUNDRED-PER-CENT SQUARES 609 


of a whole into three parts, may now be asking himself for a 
chart form which will show conveniently the division of a 
whole into four or five parts. In this case he will possibly 
have use for a hundred per cent square in which the four or 
five segments are indicated only by points and arrows or short 
lines. Thus if we are dealing with the sales of the four lines 
of a company in many different sales districts, we can combine 
them into two groups of two each and plotting each group as 
layers, we can indicate the division lines in the layers by short 
lines from the layer-division line only for its distance between 


OCONEE Ep OWE 


47%, 


NTE 


WORLD'S COAL SUPPLY 


(ESTIMATED UNMINED IN 1920) 


GRAND TOTAL - 7,460,506,000,000 TONS 
Fig. 470. Same as the Last in Circular Form. 


610 CHARTS AND GRAPHS 


these points. Such a chart reminds us of the famous “swastika” 
pattern. The chart is not of much general value, but would so 
abbreviate the rulings of the 100% square that many 100% 
squares could be superimposed or combined upon one chart 
with a visible record for all. Of course, the sizes of the areas 
would only have relative significance here, as the total areas 
for all grand totals would be the same regardless of their 
absolute values. 

Closely related to the 100% square is a special type of 
100% circle or pie-chart which has recently come into vogue. 
By means of an elaborate method of segmentation, small per- 
centages and complicated groupings of parts can be shown 
without difficulty. For it is obvious that by the simple method 


REMAINOeR of 
EUROPE 


Jewish PoPuLation of \WoRLO 
1920 Estimates 
Gaaso Terars [5,000,000 


Fig. 471. 


of segmentation of the pie-chart, in which each segment or 
part of the circular area extends from center to circumference, 
the small segments or parts become long thin attentuated 


HUNDRED-PER-CENT SQUARES 611 


areas, which are not easily labelled. The more elaborate 
method breaks the circular area into concentric rings, the width 
of each ring being particularly calculated to fit a particular 
segment in the circle, and to result in significant areas within 
the segment inside and outside of the ring. The making of a 
chart of this kind is not as easy a matter as with the simpler 
method, when all segments extend from center to circumfer- 
ence. For virtually each angular segment, or slice of the pie- 
chart is subjected to crosswise segmentation, and the ring-like 
division lines, or arcs, must be placed at distances from the 
center which correspond, not to the ratio of the parts to the 
whole of the segment, but to the square root of that ratio. 
The calculating is not easy. But the chart has very definite 
advantages for detailed and minute data when it is desirable 
to show several groupings simultaneously. There is little to 
recommend it for purposes of precise and comparative study, 
but for popular and unscientific purposes it 1s much favored. 


agriculture, forestry, and 
animal husbandry 


FARMER, ETC 


Extraction of mincsrale 


LABORER 


y LAGORER 


LABOKER 


Extraction of minerale SEMI-SKILLED WORKER 


Manufacturing and mechanieal 


SKILLEO WORKER industries 


Transportation 


Transportation 


CLERK 


Prict 


Domeotic and personal service 


Clertenl cceupations 


OCCUPATIONS OF NFULLY EMPLOYED POPULATION 
Qo ver) 


(S01 ©. r St 
(mite arce present the 3; Dla ube female workers) 


Fig. 472. A Third Classification has been Added Here by Diagonal 
Divisions and Shadings, Showing Sex. 


The 100% square or rectangle, and its many variations, are 
adaptable to a wide variety of uses. No set rules can be laid 
down to limit the various ways in which it may be applied. 


ore CHARTS AND GRAPHS 


But a very careful study of the results should always be made, 
to ascertain that one of the simpler methods in which areas 
have no especial significance would not, after all, have pro- 
duced more simple and forceful results. The danger is not 
that the ingenious chart-maker will fail to utilize all the 
possible significant features of the compound area chart, but 
that he will utilize too many of them, overcrowding his chart 
with complex details. The simpler, the better, both for research 
and publicity. And as area charts are more generally popular 
in their appeal, simplicity is a cardinal virtue. , 


Cuapter LI 
AREA-BAR-CHARTS 


It has been already laid down as a general rule that area 
charts (that is charts in which both dimensions of a charted 
area vary) are useful only when the data represented by the 
area is of less importance than the data of its factors. For 
the area chart is based upon the geometrical theorem that the 
number of units of measurement in a rectangular area is equal 
to the product of the linear units of measurement along its 
two sides or dimensions. From this it follows that we can 
always show a numerical value by an area whenever we can 
break that numerical value into two factors and can plot these 
two factors as the two dimensions of the area. 

In some cases the presence of two factors in the data or 
the fact that the data is the product of two factors, is so 
obvious as to be self-apparent. Thus the floor space of a room 
is obviously the product of its length and its breadth, and a 
chart of the room showing its dimensions and resulting area, 
could be constructed by the veriest hovice. But should we 
come to compare a number of such rooms, it would be a real 
question whether to show the dimensions of the rooms or to 
show only their total areas, that is, whether to use an area-bar 
chart or an ordinary bar-chart. If the figure for total areas 
(or square feet) is more important, we must drop these variable- 
area charts and present the data of square feet along one 
dimension only by a bar-chart. If, on the other hand, it is 
the shape or dimensions of the rooms in which we are more 
interested, then of course we should adhere to the area diagram 
and let the reader rely upon guesswork or upon appended 
data for the total area. When both aspects of the data are 
important, it would indeed be best of all to use both methods, 
outlining the shape of the room by small area diagrams and 
showing their comparative sizes by a bar-chart. This example 
of data of square measurements of a physical area excellently 

613 


614 CHARTS AND©*GRAP ES 


illustrates the fact that even data of the most obviously two- 
dimensional nature is, so far as the product or resultant is 
concerned, best shown by a one dimension chart. 

On the other hand, data which seems most clearly to be 
one-dimensional in its nature, can always, if you desire, be 
broken up into two factorial parts. When this is done and 
you regard the factorial parts or factors of the data as more 
important than the data itself, you can then proceed to show 
these factors with their products by an area-bar chart. This 
breaking up into factors, it may be remarked, can always be 
obtained by a process of division. Thus the sales of our com- 
pany in various States may be divided by population of these 
States, and so the per capita sales will be obtained. The same 
total sales in each State might also be divided by the number 
of dealers in each State and so the sales per dealer be obtained. 
Or these total sales might be divided by the similar total sales 
of the previous year, and so the percentage of increase be 
obtained. In short, the most palpably one-dimensional data 
may, by the process of division, be turned into factorial two- 
dimensional data and shown by the area chart. 

Making the chart, it is neither desirable nor commonly 
feasible to place the zero line of both axes of all the areas 
together, for this would require that they be superimposed 
upon each other. The result would be the same as if, in making 
the multiple bar-chart, we had superimposed the correspond- 
ing bars, for each lower bar would be at least partially hidden 
by the upper one and if the upper one were at any time longer 
or larger than the lower one, the lower bar would of course be 
entirely hidden. This method of superimposition is occasion- 
ally used in area charts when the difference between the com- 
pared areas along both dimensions is very great. We then 
have the effect of squares or rectangles within squares or 
rectangles, the inner one being placed at one corner of the 
outer one. The reader must then be carefully warned that 
the larger area includes the smaller one and is not alone com- 
posed of its visible portions outside of the smaller one. In 
general, the method is unsatisfactory and to be avoided. 

The proper method of showing areas to be compared is to 
arrange them side by side, so that along one of the dimensions 
only, the area will have a common zero line or base-line. 
The result then closely resembles a bar-chart, its only distinc- 
tion being that the bars which form the areas are not of a 


AREA-BAR-CHARTS 615 


constant width, as in the bar-chart, but are of varying widths, 
the variations in width showing the second factor in the data. 
And it will be seen that both the varying widths and the 
resulting areas are of secondary importance, serving to give 
the reader of the chart a general impression of the relative 
importance of the items which are described by the var- 
ious lengths of the area-bars. For as has been repeat- 
edly pointed out, the reader will have difficulty in precisely 
comparing areas of different sizes, and it is obvious that he 
will also be unable to gain exact impressions of the various 
widths. The most that can be said for the chart is that it 
gives him a precise knowledge of statistics of one factor in 
the data (as shown by the lengths of the bar areas)—in this the 
chart has all the virtues of a bar-chart—and that it also gives 


‘APRIL 1921 SALES 
100% 


APRIL 1922 
CLOTHING 135 


DRUGS eee ee [OS 


GROCERIES 


RO) 
Breese Geog 


ed I, 
HARDWARE me 


DRY GOODS 


Fig. 473. A Simple and Excellent Area Bar-chart. 


Sales of wholesale concerns in the Second Federal Reserve District in April, 
1922, compared with their sales in April, 1921. Width of bars indicates relative 
amount of goods sold.—Permission of Mr. Carl Snyder. 


him a general impression of the other factor and of the result- 
ing product of the two factors—in this the chart is an improve- 
ment upon the bar-chart. 


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AREA-BAR-CHARTS 617 


The natural arrangement of the area bars would appear 
to be in a column down the page. The statistics and labels 
or items would then also be placed in columns to the left of 
the bar area, the whole chart closely resembling the bar-chart. 
To this chart we may give the name of “‘area-bar-chart.” The 
chart is dignified, sound, and extremely illuminating. It 
requires but little more labor than the ordinary bar-chart, 
for the minor importance of the second factor and the resulting 
product, shown by widths and areas of the bars, makes it 
unnecessary for these to be plotted with extreme accuracy. 
If a‘scale for the widths be chosen small enough, the data and 
labels or items appended to the chart can be entered at fairly 
uniform distances down the chart, making for a very present- 
able appearance. The small scale upon which the widths of 
the bars are plotted deters the reader from attempting pre- 
cisely to estimate the secondary or less important factor and 
the area, while it nevertheless gives him an excellent idea of 
the relative importance of the primary data presented in bar- 
‘chart form. The chart is a direct outgrowth of the simple 
bar-chart, and in its proper place a decided improvement. 

By far the more popular form of area-bar-charts however, 
is the modification of the pipe-organ or vertical bar-chart. 
To this form, the name of “‘sky-line chart”’ is sometimes given. 
In the sky-line chart, however, it is customary to give to the 
widths of the bars, a somewhat larger scale, showing their 
variations with more precision and emphasizing differences in 
areas. Partly on account of the greater widths and partly 
for the increased spectacular effect (which is always desired 
in popular charts) the areas are placed in direct contact with 
each other, being strung out across the page in vivid resem- 
blance, let us say, to such a silhouette as the sky-line of lower 
New York City as it is first seen by a visitor from abroad. 
In this chart distinctive shadings or color tints are often desir- 
able to distinguish the areas because of their close contact 
with each other. For the same reason very narrow bars which 
could only be shown by thin vertical lines are better shown 
with narrow separating margins between the lines or in the 
form of the previously described area-bar-charts. 

In the choice of shadings in these charts, as elsewhere, 
where shadings are used, care must be taken to avoid optical 
illusions, produced by bringing together shadings of different 
color density, for it will be found that two equal areas will 


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AREA-BAR-CH ARTS 619 


appear unequal if one is much more densely shaded than the 
other. In the sky-line chart care must also be used in the 
entering of data and item labels, for the same difficulties of 
typography which were met with in the pipe-organ chart may 
be encountered here and the same considerations (discussed 
in the chapter on pipe-organ charts) in general, apply. 

In both area-bar-charts and sky-line charts, it is sometimes 
useful to show by a dotted or broken line the contour of the 
entire group if all its varying areas were combined into one 
area. This light, broken line serves to show the average or 
normal or typical phenomenon from which the individual 
areas are variations. 

When the items or stubs (the independent variable) of the 
data form an ordered numerical series, we find ourselves, in 
area-bar-charts, back to frequency curves in staircase form. 
The frequency curve in staircase form is essentially nothing 
more than a sky-line chart, generally of a certain charac- 
teristic type (the tallest area-bar being near the center and 
those at the side diminishing in height gradually until they 
vanish entirely). In the chapter on frequency curves we 
have seen that when the items form a continuous and not a 
discrete series, a smoothed curve can be plotted through the 
mid-points of the top ends of the area-bars making a frequency 
polygon. And we have seen that while the total area under 
the frequency curve truly represents the total aggregate of 
the frequency series, nevertheless the areas between any two 
ordinates under the smoothed curve are not individually equal 
to the corresponding area-bars in the staircase frequency curve 
(unless the adjoining values about any particular item happen 
to form an arithmetical series). “This inaccuracy, we have 
seen, is sometimes more than compensated for by the more 
suggestive value of the smoothed curve. 

A more complicated form of the area-bar-chart occurs when 
the areas are segmented like 100% bars to show a secondary 
classification or subdivision. This chart closely corresponds 
to the compound bar-chart, and the 100% square, differing 
from the former in the varying widths of its bars or layers, 
and from the latter in their varying lengths. It is generally 
of use in the analysis of the parts of a frequency series, either 
cumulated or simple. It is a sort of frequency band curve, in 
which the actual values are plotted on both axes, where the 
100% square, rectangle, or marble-cake (smoothed vertical 


620 CHARTS AND GRAPHS 


curves) chart projected the secondary classification only in 
percentages. The “stream chart,” too, in which the bars or 
areas are arranged on both sides of an axis, can be modified 
to present similar area features. pus 
In the particular case where the data contains two pairs, 
with one product always equal to another product, the two 
areas representing these two pairs can be pictorially shown as | 
suspended and balancing each other upon the two arms of a | 
chemist’s weighing scales or balance. The two areas must of 
course be suspended at equal distances along the lever arms of 
the balance. The pictorial representation of the balance 
suggests to the reader the equality of the two products. 


a black areas indicate weights, or counter-poises, the equilibrium of which corresponds to the “equation of exchange.” 
hese black areas !from left to right represent: 


M’, i.e., bank deposits subject to check, in billions of dollars. 


M, i.e., money in circulation in the United States (outside of the United States Treasury and the banke), in billions 
‘vf dollars. 


T, i.c., the volume of trade circulated in billions of “units” (each “unit” ‘being that quantity which could be: 
purchased for one dollar in 1909). 
Tho lever arms of the above three weights represent: 
V’, i.e., the Velocity of circulation (“activity”) of the deposits,, M’ 
V, ie. the velocity of circulation of the money, M. 
P, de., the index number, or scale of prices, at which the trade, T, is conducted. (This scale of prices is measured 
a8 8 percentage of the scale of prices of 1909.) 


Fig. 476. Balance or Counter-poise Chart with Two Factors. 


The chart illustrates Professor Fisher’s quantity theory of money, according to 
which (M’x V’)-+(M x V) =(T x P) that is, checking deposits times their velocity, 
plus money in circulation times its velocity, equals prices times volume of trade. 
‘The weights are here shown as horizontal lines and their factors as leverages.— 
Permission of Mr. Irving Fisher. 


AREA-BAR-CHARTS 621 


A more striking method of using the same idea is to show 
one of the factors of each product by a weight or bar suspended 
from the lever arm and the other by the distance along the 
lever arm between the weight and the fulcrum or point of 
balance. In this case, we have, as it were, abbreviated the 
area, merely showing its two dimensions and leaving the reader 
to imagine the area itself by projecting these two linear dimen- 
sions. The picture has little analytical value but is a powerful 
means of visualizing to the reader the mathematical relation, 


AB = XY. Professor Irving Fisher has used a series of the 


two-factor scales or balances with excellent effect in his exposi- 
tion of the quantity theory of money. 

This method can indeed be extended to show the equality 
of products not of two, but of three, factors as well. Each pair 
of “balances” or weighing “‘scales” illustrates an equation of 
the form ABC=DEF. And a series of such charts would 
illustrate a series of such equations. Since we have used the 
radial distance from the fulcrum or point of balance to the 
point of suspension to represent one factor, we can show the 
weight there suspended as a two-dimension area, the length 


ES BUION vn eZ 


9) 
Fie] ike] ike} Oo 30 40 so 60 Oo 80 90 2 
i * 


Fig. 477. A More Pictorial Form of the Preceding Chart. 
A detail of the chart in Professor Fisher’s book, “The Purchasing Power of 
Money,” in which the second factors, that is, in the chart, the weights, have 
been pictured realistically —Permission of Mr. Irving Fisher. 


622 CHARTS AND GRAPHS 


and width of which illustrate the other two factors. Such 
balance pictures need not be the same in pattern on both sides 
or areas of the balance; we may for example show the three 
factors on one side and their total product upon the other, 
using for the latter merely a horizontal bar. Thus total sales 
may be shown as a plain bar-chart, each bar being centered at 
a fixed point on one side of the balance, and per capita sales 
may be shown as an area-bar chart, positioned at the other 
end at distances corresponding to the population. The areas 
would show a long one dimension the “last year’s per capita 
sales of the quota” and along the other the present percentage 
thereof. In short the balance or weighing scales chart is but 
a form of compound area-bar chart with especial pictorial 
value for its particular relations. 

The ingenious chart-maker will be able to apply the prin- 
ciples of the area-bar chart in a wide variety of ways, always 
remembering that the widths and areas are less significant 
than the lengths and should be only used as qualifying or 
secondary information serving to evaluate or weigh the im- 
portance of the primary information shown by the length of 
the bars. In some cases, it is possible to apply these principles 
even to such well established bar-charts as the Gantt progress 
chart. A step in the direction of this qualifying evaluation 
was taken in the chapter on bar-charts when wider bars were 
recommended for total-group and sub-total bars in a bar-chart. 

There is, indeed, no reason why the area chart principles 
cannot be applied to circular graphs. And in the next chapter 
the reader will find the same principles extended to wholly 
irregular areas, such as map outlines. In general, the area 
chart like the 100% square, is essentially popular in its appeal 
and simplicity is therefore of first importance. Although every 
digression from simple linear measurement results in a loss of 
precise legibility, yet when properly used, the principles of 
areas can be made to improve a great many charts both in 
attractiveness and in instructive value. 


CuHaptTer LII 
POPULATION MAPS 


Every digression from simple linear measurements results 
in a loss of precise legibility. In the rectilinear area chart we 
have seen that the area itself was but a poor illustration of 
the values it represented and was therefore useful chiefly for 
the- sake of the general impression which it gave of relative 
importance of items or values already illustrated by lines. 
We are now about to take still another step away from simple 
linear dimensions and make use of areas of irregular outline. 
It is therefore more than ever necessary to repeat that the 
areas have little more than a qualifying or evaluating use, 
serving to give a general impression of the relative impor- 
tance of items. 

We do not sell our goods to the mountains, bill them to 
the rivers, or credit the forests with payment. Probably from 
at least a subconscious appreciation of this circumstance, 
many national distributors, advertisers, and sales-managers 
have discarded maps on which the rivers, forests or mountains 
are shown when they are studying the geographic distribution 
of their sales. The up-to-date sales manager plots his dis- 
tributing points and records his sales in a great many ways 
upon maps which carry only faint State outlines or at the most 
show the location of the larger cities. But why stop here? 
Your sales manager does not sell to square miles, acres, or 
other units of land-area measurement. He sells to human 
beings. Why should he use maps which show, not human 
beings, but square miles, that is, maps in which the areas 
indicate not the population but the land surface? Why indeed! 

The average density of the population in the United States 
proper at the last census was thirty-five persons per square 
mile. This density however varies from State to State. In 
some New England States there are more than four hundred 

623 


i 


624 CHARTS AND GRAPHS 


Feo Tie Cea cape 


quare Miles. 


fferent parts of the country. 


in di 


On this Map the Areas Represent S 


is very different 


Every Map is an Area Chart. 


Fig. 478. 
Showing that the value of farm-land, shown by the shadings, 


$25 to $50 per acre. 


EEA $50 to $75 per acre. 


C7 Less than $10 per acre, 
GM $100 to $125 per acre, 
Baa 3125 and over per acre. 


YU“. $10 to $25 per acre. 


ZZ. $25 to $100 por acre. 


POPULATION MAPS 625 


persons per square mile while in some Rocky Mountain States 
there is less than one person per square mile. 

A handful of peas in the bottom of a box can be kept in a 
small corner if you hold them with your hands, but if you 
release them, they will quickly spread most evenly over the 
bottom of the box almost like water. Imagine the population 
which is pent up in these small eastern States, suddenly being 
released like the peas in the bottom of the box, and flowing 
out over the land of the United States until its density is 
uniform throughout, that is eight persons per square mile. 
Also imagine the population as carrying its State borders with 
it so that the enormous population of the Northeastern States, 
spreading out more than half way across the continent, would 
carry the borders of the northeastern States westward and 
southward with them. (Readers of this book who live west 
of the Mississippi river or south of the Mason and Dixon line 
may omit the remainder of this chapter!) 

The result of this projection of the map of the United 
States upon a population basis rather than a land-area basis 
will be most surprising even to the most hardened travellers. 
A comparison of such a population-projection map with a land- 
area projection map will show how far the State lines have 
been shifted. From a position about one-third of the way 
across the map from the Atlantic ocean, the Mississippi river 
shifts to a position about a fifth of the way from the Pacific. 
The Rocky Mountain region becomes a narrow strip on the 
map. The Southern States shrink frightfully. But if the 
familiar outlines of the States are approximately kept in the 
new projection, the States will still be easily identified in their 
new form and you no longer have difficulty in locating im- 
portant but crowded eastern cities. 

Needless to say, the picture of sales conditions which such 
a map exhibits, will be far more valuable and useful than the 
picture upon the usual land-area basis. For in spite of a 
thorough knowledge of the various State populations, even an 
expert on population statistics will find less difhculty in visual- 
izing sales conditions as far as the real market, that is the 
population itself, is concerned. You will no longer attach 
erave importance to the far Western States which show up 
poorly on your colored scale map, for they wall no longer be 
enormous and terrifying red areas. But you will attach far 
more importance to the red color when it appears in the 


627 


POPULATION MAPS 


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ONIGNSd SLNALVd 


628 CHARIS AND GRAPHS 


eastern States. Per capita sales statistics, especially, will be 
useful on this projection. And the location of sales distribu- 
tion points, branch offices, and representatives or other sales- 
men will on this map show how evenly the market is saturated 
with your agencies rather than how regularly they are placed 
as mile-stones across the country. In short, the corrected 
areas of the States serve to give an excellent background or 
evaluation of the importance of the statistics plotted upon the 
map. 

Other conditions beside the total population can be made 
as the basis of projection of the map. If your market is best 
shown by the native white population, for example, the map 
should be projected upon not a total population basis, but 
upon the basis of this particular class of the population. If 
your market is best shown by the dealers, retailers, jobbers 
or brokers, then the map should be based either upon the 
number of distributing agencies or upon their aggregate finan- 
cial rating. For the analysis of business or economic conditions 
relative or comparable to the wheat crop, or any other form of 
produce or natural resources, a map might be called for, which 
is projected upon the basis of the average yield of this par- 
ticular crop or resources during the past years. 

The number of ways in which the map can be altered and 
projected for special purposes upon special bases is unlimited, 
but all are alike in one respect—that their areas no longer show 
physical land areas in square miles but show the actual values 
more important for the special purposes in view. This does 
not mean that a land area map is useless; on the contrary, 
there are many processes, such as shipping, railroading, and 
travelling, for which the actual land distances are important. 
It is only intended to show that where land area is not im- 
portant and some other condition is important, it is possible 
for the map to represent the values we consider important, 
whatever they be. 

The making of such special projection maps is very difficult 
and tedious and it is much better to purchase them from the 
few publishers who supply them. When you must prepare 
them yourself, perhaps the best method is to use a large sheet 
of cross-ruled paper in which the co-ordinates cut the paper into 
small squares of one-tenth of an inch or one millimeter each. 
Having before you a table of the values which you wish to 
project as areas on the map, lay out by the rule of “try, tEY, 


POPULATION MAPS 629 


try again,” the State outlines, taking care to maintain their 
familiar shapes as far as possible, but at the same time counting 
the number of small squares included in the outline and chang- 
ing the outline until the right number of small squares has been 
inscribed. For checking purposes, a planimeter (an engineer’s 
instrument which measures areas) is useful. If only rough 
outlines are required, you might find that a large supply of 
differently colored small glass balls could be counted out for 
each State, one color for each State, and quickly adjusted into 
familiar outlines and closely packed to secure the right area. 
Another short cut is to use differently colored plastocine or 
children’s modelling wax, and, having weighed the right quan- 
tities to suit your data, to mold these into familiar outlines 
and press them flat to a uniform thickness. When beads or 
wax are used, color distinctions should be maintained for the 
different States. Thin strips of paper along State borders 
may help to keep the colors from mixing. | 

The salesmanager will seize upon this map eagerly for sales 
analysis. The economist will often find it invaluable. Even 
the layman, with no charting or graphic analysis to make, 
will find it of absorbing interest. The correction which the 
map gives to our conceptions of State populations makes the 
map of real educational value, and the school geographies 
should have not only national but world maps upon such 
projections. The student will note that the principle of the 
map is the same as that of other area charts. He will recognize 
that precise estimates of the values represented by the areas 
are not possible, particularly as the shapes of the areas are 
irregular. He will see that in common with other area charts, 
the real value of the areas on the map does not lie in exact 
measurements of the values of secondary importance repre- 
sented by the areas, but in the general weighting or evaluation 
of the relative importance which is given to the data of primary 
importance plotted or recorded upon the map. 


CuaprTer LIII 
MODELS 


We have now seen used for charts, successively, the point, 
the line, and the area. The single straight line, or single system - 
of straight lines, ending at specified points, forms the bar-chart 
in its many forms; if the line is circular, the pie-chart results; 
in all of these the points are perhaps the essential feature and 
the lines and areas may be considered incidental. A series of 
points or dots connected by a line, forms a curve; the plot is 
then of several points upon a dual system of straight lines, 
called co-ordinates; the outstanding feature here is a line 
(called a curve) to which intersect points and inscribed areas 
are incidental. A series of such lines or curves may be used to 
mark off areas, forming the area chart. But all of these forms 
are limited to the use of two dimensions. A third dimension 
is not supposed to be present, and is actually negligible, being 
no more than the thickness of the layer of ink, crayon, or color 
used in making the chart. 


We now come to project points, lines and surfaces upward 
from the plane surface to get charts in which the third dimen- | 
sion itself is significant. And as may be imagined, we can 
attach importance and primary significance either to the ele- 
vated point, the elevated curve, or the elevated surface, to 
the vertical lines of elevation or the horizontal lines so elevated, 
or to one or another of the edgewise planes supporting the 
elevated plane, or to the entire volume itself. In short, 
we may use as significant any or all of the various intersecting 
lines and surfaces which make up the three-dimensional body, 
or the intersect points themselves, or the points, lines, and 
surfaces within the body. And there is something new—we 
can use the cubic content of the three-dimensional body as a 
basis of charting. This is a gain in simplicity at the cost of 
other things, and with this use of the three-dimensional body, 

630 


MODELS 631 


solid, volumetric chart, or model—call it what you will—we 
shall begin. 

The reader whose mind has leapt ahead of the diverse 
forms of area charts to speculate upon the possibilities of three- 
dimension charts will not be surprised to find the “model” 
treated as a type of graph or chart. He will realize further that 
the model stands in the same relation to flat-charts as sculp- 
ture to pictures. Just as the floor surface of a room may be 
shown by the area of a plane representation having length and 
width corresponding to the length and width of the room, 
so too the cubic content of a room may be shown by the volume 
of a solid model having the length and width of the area chart 
and the further element of height corresponding to the height 
of the room. Just as the area-chart represented by its area, 
a product of two factors shown linearly, so too the solid model 
represents by its volume a product of three factors shown 
linearly. In both cases the scales for linear measures can be 
projected only arithmetically to secure the significant repre- 
sentation of the resultant or product, shown in square units in 
the area-chart; in cubic units in the solid model. 

It is important to note that in the solid model even more 
than in the area, the representation of the resultant or product, 
though precise enough, is not easily amenable to precise esti- 
mation or comparison with other such products, for the human 
eye can even less easily judge of the relative values of two 
volumes than it can of two areas. Hence volumetric measures 
should be used, as square or surface measures, only for data in 
which the products themselves are of less importance to us 
than the factors which we will show linearly and which go to 
make up these products. 

The reader who has caught the relation between the curve- 
chart and the area-chart in the realm of two-dimension charts 
will be prepared for a similar distinction in three-dimension 
charts. Comparable to the curved-line chart would be the 
curved-surface chart; analogous to the area with square units 
of measurement, would be the solid with cubic units of meas- 
urement. Such a distinction, however, is not of great import- 
ance; in neither two or three-dimension charts is it a hard 
and fast division. In the discussion of area charts, we have 
already seen that whenever the areas are classified by, ar- 
ranged in, or dependent upon an ordered numerical series, the 
areas may be fitted together to form a curve or curves. Like- 


632 GHARTS AND GRAPHS 


re) 


wise in the three-dimensional charts, whenever the solids are 
classified by, arranged in, or dependent upon an ordered nu- 
merical series, we can fit all the solids together to form a 
curved plane. But because we shall not consider as a special 
subject the single isolated cubes or solids, we shall have little 
use for a distinction between volumetric charts (in which the 
unit primarily is cubic measurement); they could be made, but 
rarely with profit. The more complicated structure of cubes, 
their more difficult presentation and inspection, and the fact 
that their outer surfaces hide their inner transverse planes 
which are essential parts of them, make such isolated solids 
and even sets of isolated solids, of little practical value. Under 
this head we shall therefore discard all consideration of seg- 
mented cubes. The experimentally minded will be able to 
construct not only 100% cubes but even sets of several cubes 
and solids of various dimensions, which latter he will be able 
to compare by means of either or all of their three linear di- 
mensions, their three areas in square measurement, or their 
one volume in cubic measurement. For the comparison of 
different buildings, engines, machines or other physical equip- 
ment or structures, such models, or small replicas may indeed 
be useful. But apart from miniature replicas of actual phy- 
sical objects (in three dimensions) there would be little use 
for such isolated models or sets of isolated models. For math- 
ematical statistics, the individual factors are better compared 
in separate sets, in bar-charts or curves, and the products are 
likewise better charted by linear measures in bar-charts or 
curves. 

Just as the area, measured in square units, is not significant 
in all two-dimension charts, so too, the volume, or cubic con- 
tent is not always significant in three-dimension charts. For 
the co-ordinates in a curve-chart, for example, need not have 
any factorial inter-relation; this is the case of historical curves; 
we do not multiply the phenomenon by its date of occurrence 
to secure a significant product. Likewise the three systems of 
co-ordinates used in projecting a solid, need have no factorial 
inter-relation which has a meaning for us; in such cases the 
three axes and sets of co-ordinates are merely convenient plot- 
ting devices which enable us to distinguish three different 
variables in our data, and to study their mutual relations 
and behavior. At other times, we may detect a distinctly fac- 
torial relation between these variables, and then, of course, 


MODELS 633 


we can identify the product with the cubic content or volume 
of the chart. 

The most convenient division of three-dimension charts, 
however, and the one which we shall here follow, is the divi- 
sion between three-dimension charts in general, and one par- 
ticular kind of three-dimension chart in which the first two 
dimensions are used to mark off geographical relations. To 
all other models and three-dimension charts we give the name 
of frequency surfaces; in these the first two dimensions have 
for their scales, any other numerical series, either historical 
or frequency. But when the numerical series indicates lati- 
tude and longitude upon the earth’s surface, or any geograph- 
ical co-ordinates, we call the chart a map and because of great 
practical use which is made of maps, we shall consider them 
in a separate chapter. As has already been pointed out, the 
isolated solid or series of separate solids, which is comparable 
to the bar-chart when the latter cannot be converted into a 
curve, that is to say, which is not classified by, arranged in, 
or made dependent upon an ordered numerical series and so 
cannot be joined into a curved surface, will not be discussed at 
all. And before proceeding to the consideration of the two 
types of three-dimension charts we shall first examine in the 
next chapter, the methods by which the third dimension can 
be shown. 


CuarptTer LIV 


THE THIRD DIMENSION 


There are three ways of preparing stereographs, that is, 
three-dimensional charts, which may be called respectively, 
the model, the axonometric chart, and the orthographic chart. 
The first requires three dimensions physically in space; the 
second illustrates three dimensions in a two-dimension plane; 
the third shows two dimensions faithfully and seeks to repre- 
sent the third dimension by some trick of symbols. 

If we elect to use the model, it may be either solid or col- 
lapsible. To the solid model, actually built up in three di- 
mensions, there is of course little structural difficulty. A solid 


From ‘‘The Construction of Graphical Charts," by John B, Peddle, published by McGraw-Hill 
Book Co., N. Y. 


Fig. 481. A Plaster of Paris Model. 


Model showing the relation between heat units per hour per brake horse-power, 
compression pressure, and volume of gas mixture for a gas engine, 


634 


THE THIRD DIMENSION 635 


model can be made of wood, or of many layers of cardboard 
or corrugated paper, and can also be moulded of plaster of Paris, 
of wax, paper-pulp, or other material. Detailed instructions 
for the modelling of curved planes will be found in the fol- 
lowing chapter. Such solid models are of course cumbersome 
and unhandy, difficult to file away, or to carry about from 
place to place; and would seem justified only in the case of 
extremely important data. Moreover, and this is important, 
the making of such solid models requires a great deal of time 
and trouble, the equipment not being generally immediately 
available for them in the average statistical office. 


Special varieties of the solid model will also be described in 
the immediately following chapters.! These consist of small 
forests of vertical wires, or wooden sticks, placed far enough 
apart to allow any individual wire or stick to be inspected. 
This ‘is in some respects the model par excellence; it admits 
of segmentation, for each wire or stick can be differently 
colored through parts of its own length. And most unusual 
of all, several models can be combined by placing their wires 
or sticks side by side with distinctive coloration. The colora- 
tion of wires is-usually achieved by stringing colored beads 
upon them. The forest model is a more or less laborious 
affair to construct, but for sufficiently important facts it is of 
ample merit to justify its use. 


Collapsible models, as the name suggests, are so made that 
they may be folded up or expanded at will. When folded, they 
lie flat upon a single sheet of paper and can be easily carried 
about or filed away as sheets or folders. When expanded to 
occupy three physical dimensions in space, they can be stood 
up like blocks or other solids. There are three types of col- 
lapsible models, with respect to the mechanics of their opera- 
tion. The first type telescopes by means of co-planar hinges, 
at right angles to each other, like the folds of a pair of bellows 
or the hood of a folding camera. The second type collapses 
side-wise upon hinges which are all parallel to each other, like 
the partitions in a pasteboard egg-box. The third type never 
opens out fully for all parts of the model are hinged together 
like the leaves in a book. 


1See also, Brinton, Willard C., Graphic Methods for Presenting Facts, Engineering 
Magazine Co., New York. Peddle, John B., Construction of Graphical Charts, McGraw- 
Hill Book Co. 


636 CHARTS AND GRAPHS 


In the telescopic model all three dimensions are physically 
represented by materials in the structure of the chart and 
telescoping is only possible by buckling up of these materials 
across one of the three dimensions. Because of this buckling 
of materials the telescopic model is not easily made or oper- 
ated, and is generally inferior to the other collapsible models. 

In the second type, collapsing side-wise, because one of the 
physical dimensions is not represented by any structural 
material in the chart, the materials lying on the other two 
physical planes can be made to fall together as easily as a 


From ‘The Construction of Graphical Charts,’ by John B. Peddle, by permission. 
Fig. 482. Collapsible Model. 


house of cards. If the hinges are on edge the chart folds out 
to right or left. In this case it is most convenient to make the 
chart of intersecting slitted sheets, the sheets to stand in one 
direction having vertical slits or key-ways, halfway up from 
the bottoms at the points where the cross-wise sheets intersect, 
the latter sheets having corresponding slits down through their 
upper halves; and the two sets fitting together, as has been 
said, like the partitions of the egg-box in which you buy a 
dozen eggs. Obviously all sheets should be of stiff material, 


rigid enough not to fall over, strong enough not to tear beyond 
the slits. 


ee 


THE THIRD DIMENSION 637 


When this type is made with horizontal hinges, only one 
set of vertical sheets is used and these are all hinged, parallel 
to each other, upon a single horizontal sheet of pasteboard. 
That the various hinged sheets may act as one and maintain 
their parallelism at all times, short tie-hinges or keys and key- 
ways are used across or near their tops. It is convenient to 
mount this chart in a heavy pressboard folder of the standard 
vertical filing type, with the back of the chart attached to one 
half of the folder and its base to the other so that the chart 
is always safely housed. Such a model opens out like the more 
elaborate, familiar valentines and works on the same principle. 
Similarly the forest model already mentioned can be made 
collapsible, paper being used in the place of wires or wooden 
sticks. This type is perhaps the best of the precise collapsible 
models. 

The third type of collapsible model has no structural ma- 
terials for two of the three perpendicular planes in the three 
dimensions, and, as has been said, never opens out fully. Its 
single system of plane surfaces, which should be parallel, 
radiates from a common hinge like the leaves of a book. It 
is in fact no more than a series of two-dimension charts care- 
fully bound together to secure perfect “registration,” one with 
another. . But frequently it is sufficient for the study and 
analysis of the data; and the fact that it is more easily made, 
and handled, and suffers less from wear and tear, strongly 
recommends it. Moreover, transparent or semi-transparent 
paper can safely be used for this chart, enabling the reader to 
note more easily changes in its various parts. 

This last form of collapsible chart is also readily susceptible 
to commercial publication. In German schools and colleges 
such diagrams are sometimes used for the study of parts of 
complicated machinery. The student no longer needs to have a 
physical model of the machine before him, but can fold back 
the diagram of each part to inspect the diagram of the parts 
inside it. In medical schools such methods are sometimes used 
for the illustration of anatomical studies, the first view show- 
ing the outer skin; the second, the underlying nerves; the third, 
the muscles; the fourth, the inner organs, with perhaps minor 
diagrams or part pages folding back for each of these, to show 
the internal structure of these organs; the fifth large sheet 
(seen by folding back the fourth sheet which carried these 
minor books or sets of small pages upon it) showing the bones; 


638 CHARTS AND GRAPHS 


the sixth, the rear wall of the muscles, nerves or skin again. 
The same method of presentation has been effectively used in’ 
costume studies for the stage (and in children’s toys), to show 
upon a top sheet the over-garments and outdoor costume; 
upon second sheets, the ordinary or house garments; and upon 
third sheets, the undergarments, for various national or 
period costumes. There would seem to be no reason why the 
same method cannot be used to present mathematical data 
in similarly constructed and hinged charts. For these books 
or series of superimposed charts, as for the collapsible models, 
a strong cover is desirable to protect the parts and the best 
cover is generally, as has been said, a vertical filing pressboard 
folder. 

All of these model-charts have a bulkiness, when solid, with 
an added element of flimsiness, when collapsible, that militates 
against their general usefulness. Moreover, in their prepara- 
tion they are costly of time, and often seem to call for ma- 
terial, not readily available in the average office. Except in 
the last form, when transparent paper can be used, or in other 
forms when transparent celluloid is used, or in the solid model 
when glass is used, these charts have the disadvantage that 
one part of the chart hides other parts; and diagonal lines, 
curves, or planes cannot be readily run through the chart to 
give interpolated readings. These disadvantages all disappear 
in the next form of three dimensional chart, which we shall 
now consider. 

The axonometric chart is one in which distances are meas- 
ured along three axes which have been represented by lines 
within a single plane surface. Every photograph and every 
picture of physical objects is such a chart. In paintings, 
drawings, and pictures of all kinds, perspective is generally 
introduced, to make the more distant objects smaller, that 
they may appear to be of the same size. Perspective requires 
that really parallel lines be shown as converging toward a 
non-existant vanishing point; it makes a fixed scale for any 
axis useless and greatly increases the difficulties of drawing. 
For statistical charts, therefore, we omit perspective, sacrificing 
thereby some of the realistic appearance of our picture, but 
giving it constant scales which make charting easy and read- 
ing accurate, 


a 


THE THIRD DIMENSION 639 


_ The most commonly-used axonometric chart is the one 
with isometric rulings.2 Isometric rulings are those in which 
the three axes of the chart intersect to form equal angles of 60 
degrees with each other. Any other angles can be used but 
these isometric angles are most convenient and satisfactory, 
as they afford the fullest detail along each axis with the sharp- 
est possible intersections of all ‘co-ordinates. One of the axes 
is vertical for the up and down dimension of the chart; the other 
two, at 60° to this or 30° to the horizontal, represent the two 
surface dimensions of the base of the chart. 

Isometric drawings are so very useful and so easily made 
that they should be adopted whenever possible for three- 
dimension charts. The isometric co-ordinates can be left upon 
the finished charts to facilitate interpolation and estimates of 
the values plotted linearly along them by the reader of the 
chart, for the rulings serve to connect the points plotted, with 
their scales, in the same way that the Cartesian co-ordinates, 
that is, the ordinates and abscissae, connect points upon a 
curve with the scales of the curve-chart. 

But often the effect of many co-ordinate rulings is so con- 
fusing in the finished chart that it is preferable to wipe out all 
the co-ordinates save those which have been calibrated with 
scales and those along which curves or lines have been plotted. 
When the isometric co-ordinates are to be omitted in this way 
from the final chart, it is better to rule the co-ordinates in the 
first place on blank paper in pencil, as they can then be most 
easily removed. The commercially-ruled paper is printed in 
ink, generally green or red, and the lines will be reproduced 
in photographs unless wiped out with Chinese white. When 
the drawings are to be traced, of course it is very easy to omit 
the co-ordinates. But isometric co-ordinates are so easily pre- 
pared in pencil that for most purposes this is sufficient. | _. | 

The scales for the isometric chart axes can be varied, of 
course, at will; and the student will naturally seek to adjust 


‘them somewhat to the ranges of the variables plotted. But 


when all dimensions, or even the two surface dimensions, are 
used to present commensurable values, and the commensur- 
able nature of these is manifest, the use of different scale- 
moduli or units of distance along the different axes, may 
result in an awkward unnatural appearance to the chart 


2See also, Professor Guido Marx in the American Machinist Vol, 31, Part 2, p. 
701, and Haskell, Allan C., How to Make and Use Graphic Charts. 


640 - CHARTS AND GRAPHS 


unless the angles of the axes are altered and the isometric 
principle departed from. It then becomes advisable to select 
other angles, which restore the natural appearance by swinging 
the whole chart about to one side or the other. The first 
consideration is, of course, the range of the variables; the 
second, the relative detail with which the variations should be 
shown. From these the total length of the chart along either 
axis can be determined and from the relative detail alone, that 
is the size of the unit distance or modulus, the proper angles 
of the axes should be determined. When it is desired to vary 
the angles for these purposes, the ready-made isometric paper 
of course cannot be used and the co-ordinates must all be 
especially drawn, 


per 


Heat Onits 
Cm. 


o 
” 
Pat 
ev 
A, 
cS) 
v 
na 


Energy Loss, 


Energy Loss, Meter-kgm.per Sec. per Sq. Cm. 


From ‘‘The Construction of Graphical Charts,”’ by John B. Peddle. 
Fig. 483. An Axonometric Chart (Not Isometric). 


Chart showing relation between journal-bearing temperature, surface velocity, 
and heat generated, etc. 


It is a distinct limitation of all drawings of solid objects, 
including isometric drawings, that they show the object from 


—— se  St—t— 


THE THIRD DIMENSION 641 


Ratio of scale-moduli or units of Tangents of angles formed 
length along the three axes. by the right and left-hand 
axes with the vertical axis. 
ee eae ile Van watz Lm gaan Nini yiice Stace oa 
Left-hand Vertical Right-hand Left-hand Right-hand 
axis axis axis axis axis 
mz ion my mx (z) (x) 
1 it tan 60° tan 60° 
(Isome|tric ruling) 
8:7 


8: 
8: 
2 
Se 
ite 


From John B, Peddle, ‘Construction of Graphical Charts." 
Fig. 484. Instructions for Axonometric Chart Scales. 


one view-point only. You cannot turn the drawing over for 
different views of the same object, as you could turn the actual 
object itself about in your hand. It is therefore necessary to 
arrange the scales of the axonometric drawing carefully to 
show the best possible view of the object. High points in the 
foreground will hide or obscure lower points behind them. The 
scales should therefore be so arranged that the high points will 
be set as much as possible in the background, and the fore- 
ground be devoted to low points; or that sufficient distance 
be allowed behind a peak in the foreground to enable us to 
show low points behind it. If, by reversing the direction of 
one scale, you can secure this result, the scale should be re- 
versed; for the result of reversing a diagonal scale is the same 
as giving the object itself a quarter-turn in your hand. The 
reversal of the other diagonal scale is the same as giving the. 
object itself a quarter-turn in the opposite direction; and the 
reversal of both axes gives a half-turn to the object itself, 
showing its rear face. In general, the best position can be 
found by a little experimenting and can, with a little skill, 
be determined in advance from an inspection of the data. 

It is, however, an advantage of the axonometric chart, not 
shared either by the model or the orthograph, that interpola- 
tion can be most easily accomplished upon it. For the axon- 
ometric chart can be made, if we wish, to show all sides of the 
chart at once, merely by using points or dotted lines for the 
parts which are apparently hidden from view. And even if 
these parts are not indicated, the scales and axes still remain, 
from which we can on the finished drawing drop parallels to 
any desired point and take a reading. Such interpolation 


642 CHARTS AND GRAPHS 


can be taken from straight lines in the manner just mentioned 
or from rounded contour lines drawn in from various observa- 
tions, according to the nature of the problem. The axon- 
ometric chart, and in particular the isometric one, 1s for most 
purposes the most satisfactory method of charting three 
dimensions. 

A feature which belongs both to models and axonometric 
charts is that they may present either staircase or smoothed 
surfaces. This is obvious enough from the fact that both are 
but series of curves; and curves, as you know, can be in either 
form. Moreover, since the curved surface or three-dimension 
chart is an interlocking of two such series of curves, one along 
and the other across the surface, it is obvious that the same 
surface may even be smoothed along one axis and staircased 
across the other, as well as smoothed or staircased on both. 
These possibilities are not generally open to the charts to. 
which we are coming, the third type of three-dimensional 
chart; in the latter the best that can ordinarily be done toward 
smoothing is to indicate contour lines, or lines of equal value, 
about each peak and valley—a lateral, if you will, rather than 
vertical method of smoothing. 

The third method of presenting three-dimension charts 
(they can hardly be called stereographs in this case) is the 
orthographic chart. In this, as already mentioned, two of 
the dimensions are precisely shown, exactly as in a two-dimen- 
sion chart, and the third dimension is indicated by symbols. 
In its very simplest form, numbers alone represent the third 
dimension, the numbers being for this purpose considered as 
symbols. But the graphic quality of a number is limited,— 
consisting wholly of the number of digits in the number itself, 
—and this is not usually sufficiently detailed or legible. We 
are therefore prone to seek other symbols to which we can 
attach numerical values and which we can explain to the 
reader of the chart in an appended “key” which corresponds 
to the scale along an axis. And since no symbols have yet 
been found which ; are capable of the infinitessimal graduations 
which a scale affords and are at the same time as easily read 
with accuracy, we are obliged to restrict ourselves to a few 
distinctive symbols. These we use not only for certain set 
values, but also for all values nearest thereto, establishing in 
this way groups or intervals along the range of variation and 
using these symbols for all values within each group. In short, 


THE THIRD DIMENSION ae 


the use of symbols involves a loss of detail in the presentation 
of the third dimension. 

Two considerations govern the use of the symbols. The 
first is that the groups to which the symbols are attached 
should be carefully chosen. This consideration is precisely 
the same as applied to the formation of frequency series.3 
The groups should if possible contain any round numbers or 
bunching-up spots near their centers. The intervals, or 
group-limits, should be regular and uniform, if the distribution 
appears arithmetical, and as nearly as possible to equal geo- 
metric intervals if the distribution appears to be logarithmic. 
These are obvious principles which the student will soon dis- 
cover for himself. The one thing of consequence is that we 
should not, as we may often be tempted to do, divide the series 
into groups with equal frequencies. Such a practise is con- 
fusing and deceiving to the chart-reader and has but limited 
meaning. 

The other consideration is that the symbols should be such 
as form a natural series in themselves, just as if they were 
numbered. This gradation of the scale of symbols should be 
such that it is obvious to the chart-reader—the more obvious 
we make it, the better is our representation of the missing 
third dimension. The reader of the chart should be able to 
see at a glance the order in which the symbols fall and there- 
after should not need to refer to the key again. Indeed, in 
those few portions of the chart where the extreme symbols 
are used, it is no bad plan to label the chart itself, right 
through the symbol, with the words “High” or “Low,” or with 
similar words. When this is done, the chart may be called 
self-contained and complete and the reader need only refer 
to the key for the numerical equivalents. Needless to say, 
however, the key should always be attached to the chart, 
showing the symbols and stating the limits of the ranges they 
represent. 

The position of the symbols upon the chart is of course 
dictated by the two independent variables in the data, the 
plot of which occupies the two co-ordinate dimensions of the 
chart. But since the dependent variable can only be shown 
approximately by symbols, not precisely, there is opportunity 
for two different methods in marking off the parts of the chart 


+See Chapter X XVII, pages 312-313, 


644 CHARTS AND GRAPHS 


to be symbolized. In the first place, we may take these parts 
precisely as we find them in the data. But such a method leaves 
to the chance boundaries of the data the question of what 
approximate values shall be shown by symbols for any par- 
ticular spot. And as a result the most abrupt transitions of 
values may take place between two parts of the chart appear- 
ing side by side. Where each part is inherently homogeneous 
throughout, the resulting chart, though much confused, 1s 
nevertheless accurate. 

But where we have reason to believe that the change from 
one spot on the chart to another is more or less gradual, we 
are justified in trying to smooth out the steps between parts 
of the chart, so that all intermediate symbols appear between 
any two non-successive ones; in other words, so that the change 
from one part of the chart to another is as smooth as the few 
symbols and the given data will allow us to make it. This 
results in a much less confusing picture, and in the circum- 
stances prescribed, a more significant one. It is the old dis- 
tinction between a staircase and a smoothed curve, with all 
the attendant details of loss of absolute accuracy over given 
areas and greater significance. Only in the three-dimension 
chart of the kind we are considering, the process is called 
“zoning” and the lines which bound the zones are called con- 
tour lines.4 


This is the orthographic chart proper, familiar enough in 
weather topography, where the lines of equal barometric 
pressure are called isobars; and the lines of equal temperature, 
isotherms. But to apply those principles to the chart, what- 
ever it be, is often a difficult task. When data is so scattered 
that many contour lines must be interpolated between two 
known points, the element of “guess”? becomes large, different 
chart-makers will often connect zones differently and it becomes 
often a decidedly hazardous proceeding. In such cases the 
method need not be employed, or if used, the interpolated 
zones may be made disproportionately narrow to limit the 
possible error as much as possible. 


Many and various are the kinds of symbols which may be 
used. One of these is so close to the mere number itself, which 
we have already mentioned, that it may be disposed of first. 
This method consists of the use of bars, areas or circles in the 


*Contour-lines may be considered a form of superimposed cross-sections. 


4 


THE THIRD DIMENSION Gs 


place of the numbers. These all have the possibility of infinite 
gradation, just as have numbers themselves, and so form an 
exception to the considerations just laid down for symbols, 
and require no key. They are not wholly satisfactory, however, 
for they involve as essential the use of one or both of the 
dimensions already given on the chart to other variables. 
Hence the symbol cannot be evenly spread over the entire 
part of the chart to which it applies and when parts are small 
and symbols large, the symbol will cover parts to which it 
does not belong when perhaps on the same chart, other parts 
are large and have very small symbols, the symbol is likely 
to be lost and is not fully graphic. 


Permission of Country Gentleman. 


Fig. 485. The Wrong Way. 


If it is desired to use such symbols as these graphs-within- 
graphs, then surely the symbol should not be measured in 
areas, such as the squares, triangles, stars, or circles, which 
one so often sees used for these purposes. [For the comparison 
between a large circle and a small one, or a large star and a 
small one, cannot be accurately made by. the reader. It is 
much better to use bars or segments of circles, all requiring 
linear measurement only. For in this case the reader can be 
trusted to arrive at approximately accurate conclusions, in 
spite of the fact that the bars have not been aligned at one 
end. The one case in which areas (in square units) should be 


646 CHARTS AND GRAPHS 


used seems to be the case in which the variations symbolized 
seem to increase, not in arithmetical or geometrical series, but 
in a series of squares and the special square-root projection 
is desired for the scale of the symbolized function or third 
dimension. 


“e 
@ 200,000 cattle. 
@ 150,000 to 200,000 cattle. ®@ 

‘ OOO) 
@ 100,000 to 150,090 cattle, \ * @ é ‘ 
@ 60,000 to 100,500 cattle. Se. ‘ “ose fir, 
© Less than 60,000 cattle, NN — 
The heavy lines (ee) chow geographic divisions, \ H 
~J 


From U.S. Census. 
Fig. 486. Somewhat Better. 

A logical development of this is the use of many dots or 
small circles in the place of one big one for each symbol. By 
counting dots the reader can get the exact value of the variable 
plotted in each part of the chart, unless the dots are allowed to 
become so numerous that they cannot be counted. This last 
trick, of putting in a great many dots for a single symbol, is 
unfortunately a popular, though pernicious, practise—it is as 
if che chart-maker were saying to the chart-reader, “‘I have gone 
to a great deal of useless labor in putting in all these dots, 
now you can waste your time counting them.” 

When the parts of the chart are of uniform size and the dots 
are evenly distributed within each part, the dot system is 
excellent; for the density of the dots is a graphic guide in itself; 
but when the parts to be labelled with symbols are not of even 
size, and no significant relation holds between the size of the 
parts and their symbols, then unhappily the dot-system falls 
down again: for a few dots in a small part of the chart will be 
more impressive than many in a large part. 


THE THIRD DIMENSION 647 


The poly-dot symbol leads us logically to the frank use of 
shadings, regardless of the number of dots, lines, or other 
markings in a shading. And this is ordinarily the most satis- 
factory of all symbols. For a few different kinds of shadings 
can be easily devised, which are not only mutually distinct, 
but also have a definite order of intensity, ranging from solid 
white to solid black. These shadings can be laid on with 
successive hatchings and cross-hatchings, with a section-liner 
or tee-square. hey require least work if the shadings are so 
chosen that each successive symbol has only an added system 
of lines or other markings to distinguish it from the previous 
one, for in this case all of the lesser shadings can be put on in 
the course of making the extreme shading. 

It is an advantage of the cross-hatching symbols that they 
do not, as a rule, interfere with lettering which may be also 
wanted on the various parts of the chart; the lettering can be 
read through all but the solid black or very dark symbols, 
and in the latter cases, the symbol can be omitted immediately 
about the lettering. It is also an advantage of the hatched 
symbols that they can be reproduced, in common with the 
methods already described, in a variety of ways, including the 
line-cut for printing and the mimeograph for offsetting, and the 
blue-print or Van Dyke print for copying. 

Closely akin to the hatched symbol, is the solid shade or 
tint, the shading proper ranging from pure white through 
various grays (made by mixing those two ever-present visitors, 
India ink and Chinese white) to solid black. These tints 
could, of course, be infinitesimally graduated to suit precisely 
the values plotted, but this would require unnecessary work 
and could not, through optical illusions, be correctly read by 
the chart-reader. It is therefore sufficient to use some five or 
six equally different shades which can be easily distinguished 
on a key. The method of solid shades is not, however, gener- 
ally of enough benefit to warrant its use. It requires some 
labor in mixing, the liquid may warp the paper or run, the 
symbols are never so distinguishable as hatched patterns, and 
the resulting chart cannot be reproduced by line-cut, or any 
other method except photo-engraving or photostat.> 

The most important and satisfactory symbols possible are 
solid colors. These can be made very pale so that lettering 


5 The “Ben Day process” can be used_on a line-cut to make it slightly resemble a 
halftone. See Appendix C. 


648 CHARTS AND GRAPHS 


or even a separate scheme of cross-hatching can show through 
them. Transparent inks and water colors, used in color photo- 
graphs, can be used. Better than ink or water colors, however, 
are wax crayons, the cheaper and waxier the better. They do 
not require careful mixing, lay on in even density for each 
color and do not wrinkle the paper. After the wax color has 
been heavily applied, the chart should be carefully scraped 
with a sharp knife (a safety razor blade is excellent for the 
purpose) and all the surplus wax removed. A pale tint will 
remain on the paper, through which typewriting or other letters 
or chart drawings (if previously applied) will show clearly. 
The important thing about the scale for colors 1s that the colors 
should be in what is called chromatic sequence, the order of 
the colors in that of a rainbow or spectrum. Using five colors, 
red, orange, yellow, yellow-green and blue-green will be found 
excellent. For more colors a dark red and a blue can be 
added. But five clusters or symbols are ordinarily sufficient, 
yellow representing ‘‘average or normal’; orange, “‘poor’’; red, 
bad”’; yellow-green “fair”; and blue-green, “‘good.” 

Some writers have advised an arrangement of colors by 
what they call optical density and have attempted to deter- 
mine a color density sequence. ‘These efforts have naturally 
and necessarily failed—such a scheme, even if it could possibly 
be standardized for different inks, papers, and color combina- 
‘tions, would only result in conglomeration through which the 
reader would need the constant assistance of the key. The 
only disadvantage to colors is their varying photographic 
reproducing powers, blue disappearing wholly and turning 
white, while red becomes black. A careful chromatic scale 
through the colors from red to blue will photograph as a fairly 
uniform scale of grays from white to black. But the chief 
advantage of the chromatic arrangement of the colors is their 
logical significance. The reader of the chart if he be not color- 
blind, need only knowthat blue is good and red is bad and is 
at once prepared to interpret all the intermediate colors. 

You will see that colors, shadings, and even figures alone, 
constitute a dimension in themselves upon the chart. And in 
almost all instances where models are made for three-dimension 
statistics, the same could be charted upon a two-dimension 
chart with the use of colors, tints and cross-hatchings in the 
place of the third dimension. That the symbolical presenta- 
tion of the third dimension is more a series of approximations, 


‘ 
| 
h 


THE THIRD DIMENSION 649 


no precise graduations being possible, has already been ex- 
plained. But in most cases these approximations are entirely 
sufficient. 

When a more graphic or vivid representation of the same 
three-dimensional data is desired, with perhaps more precise 
presentation of the exact values of the third dimension, the 
isometric or other axonometric drawing must be used. But in 
this a part of the data may be hidden behind peaks. If this is 
the case and all parts of the data must be visible, then of 
course you must fall back upon the three-dimension model 
itself, either in collapsible or rigid form. The great time con- 
sumed in the making of these and the inconvenience in handling 
them makes the three-dimension model, rigid or collapsible, 
justified only in the case of very important statistics, but the 
ease and convenience of colors and isometric projections make 
them of very wide general usefulness. 


CHAPTER EW 
FREQUENCY SURFACES 


- The double frequency series is a type of data which can 
invariably be recognized by the form of its tabulation. It is 
composed of several columns of figures which have common 
stubs, and in which the values of the stubs and the values of 
the column headings both form mathematical variables. In 
the body of the table, that is, at the intersections of rows and 
columns, appear the values of the functions or, in a loose sense, _ 
Pe. the dependent variable. The general form is the same as the 

i table of original data for the 100% square, already discussed, 


WET AND DRY MONTHS 


Summary of the mmber of times each month has been first, second, 
third, etc, in order of humidity during the years 1868 to 1905 — 
38 yoars in all. Taken from flow of Croton River, N.Y. at dan. 


: 

5 Jan. |rob. Mar. Apre May» Hume July Auge Bope Pct» Nov. |Dece 

; 

7 Wettest 5 68/18 ze 

i Second x ies 12 ralasieh Tees mae 

; a Ca es 

; she cals Be ee 

f une 4:57 eh Core ee eae 

p oe a dV ss Pees 

see ; me 
sett ae ae ee a so) 
= BADER 
ce Ze a 


CCC 
= A 


Fig. 487. 


but the data is not turned into percentages or products of per- 
centages as in that case. It is only necessary that the stub 
650 


FREQUENCY SURFACES 651 


and column heading figures, that is, the two independent 


variables, each form ordered numerical series. Whenever this . 


is the case, the values of the function (that is, the detail 
figures of the double frequency series, shown in the body of 
the table) can be charted in a third dimension. If you think 
of pins stuck into the table upon each figure in the body of 
the table, the pins representing the figures by their heights, 
you will see at once how this is done. 

Let us assume that we are standing in a room of rectangular 
or square floor shape. Let us mark off a pattern upon the floor 
of this room, a pattern of criss-cross or co-ordinate lines, 
calling those which run the length of the room the x-abscissae 
and those which run across the room the y-abscissae. At the 
many intersections of these two sets of abscissae, let us drive 
tacks into the floor and into the ceiling overhead and run 
strings vertically from the floor to the ceiling. Let us call 
these vertical lines the z-ordinates. At equal distances up 
these ordinates, or vertical strings, let us fasten horizontal 
strings from ordinate to ordinate above both the x and the y 
abscissae so as to produce a complete net work of crossed lines 
in the room, which would present the appearance of plain co- 
ordinate rulings when seen from above or from either side. Let 
us assume that in some mysterious way we can wander about 
this room freely without becoming entangled in the net work 
of strings. 

To the x-axis, or distance down the length of the room, 
let us give the values of time, letting’ the first unit of distance 
represent one year, the second the next, the third the following 
year and so on, so that along the length of the room we have, 
on the x-axis, a scale of years. And at each year we notice the 
cross-wise lines and the vertical lines are repeated to form a 
co-ordinately ruled plane perpendicular to the x-axis. To the 
cross-wise distances of the floor along the y-axis let us give, 
for.example, a scale calibrated to tens of dollars, and running, 
let us say, from zero to two hundred dollars, there being twenty 
cross-wise divisions of measurement. To the vertical lines or 
the ordinates parallel to the z-axis, let us give a scale calibrated 
in hundreds from zero to one thousand, there being ten vertical 
units of measurement. In short, to each of the three dimen- 
sions or axes of the cubic volume of the room, we can attach 
scales of calibrations similar to the scales in ordinary curve- 
charts along its two dimensions or axes. 


652 CHARTS AND GRAF i0e 
Returning to the end of the room to the vertical plane cut- 
ting across the room at right angles to the x-axis and inter- 
secting the x-axis itself at the point of the first year, let us 
chart upon this plane a frequency curve showing the number 
of sales of various sizes for the concern whose business we are 
analyzing for the year indicated on the x-axis. Then on the | 
| 


next plane, intersecting the x-axis, the point on its scale for 
the next year, let us plot a similar frequency curve for the next 
year. And so through all the planes, let us plot frequency 
curves, one for each year upon the plane intersecting the point 
of that year upon the x-axis. Let us plot these curves by 
attaching red strings to the network of strings which makes 
up the planes. 

Having completed the series of curves for all the years, 
we can step off and look at the result. The curves are likely 
to show great similarity with but slight changes in their exact 
positions from year to year. These changes of the positions 
of the red-string curves show us the changing nature of the 
sizes of sales made by the house. ‘The series of red strings 
seems to outline a billowy blanket or irregular curved plane. 
If through the series of curves we should attach similar red 
strings running lengthwise down the room, connecting corre- 
sponding points upon the curves, this blanket or suspended 
irregular surface would be more visible. Examining any one 
of these new connecting strings, we would find that each one 
of them forms the curve of the historical changes in the number 
of sales of each size through the various years. In short, the 
blanket could have been made by plotting the various historical 
curves along the room rather than the frequency curves across 
the room. The blanket itself is in fact nothing but a historical 
projection, carrying a frequency curve through a number of 
years, and yet exhibiting all its points at any point of time. 

It should be remarked that the name “double frequency 
series” given to this type of data is a loose one, used to describe 
both truly double-frequency series and _historical-frequency 
series. The curved plane which we have just plotted obviously 
represents a historical-frequency series, that is, the data of a 
frequency series carried through a number of periods or points 
of time. The double-frequency series proper is similar in all 
respects, except that time is not one of the independent 
variables. In the double frequency series proper, the data of 
a frequency series is carried through a number of changing 


FREQUENCY SURFACES 653 


conditions or classifications which, like time, form a connected 
or variable series. The distinction between the historical 
frequency series and the double frequency series is not im- 
portant, either in computing or charting. 


7 


Hh 

Posy Nal 
<<EX) / 
—— Sef ay 


2 = Sf IX 
SSS SSS 


From “Theory of Statistics,” by G. U. Yule (fourth edition), published by J. B. Lippincott. 
Fig. 488. Smoothed Frequency Surface. 
Showing the correlation between the height of fathers and sons. 


As you will see, the name “‘curve”’ is really a misnomer for 
this form of chart, for the chart does not merely exhibit a 
curved line, but exhibits a series of curved lines forming a 
curved plane. The word “surface” is ordinarily used for this 
form of chart, though it might also in a loose sense be called a 
curve. It is not always true that the succession of curved lines 
or true curves can be joined to form smoothed surfaces. It 
may be that the phenomenon requires that the joining be made 
in staircase fashion. The same consideration applies to the 
smoothing of the surface between plotted points as applied in 
the smoothing of the ordinary curve. It may even be desirable 
to smooth the curves along one axis and leave them in staircase 
form across on the other axis. ‘The familiar and most useful 
forms of the curved surface are, however, the smoothed sur- 


654 CHARTS AND GRAPHS 


face (smoothed along both independent variable axes), and the 
double-frequency polygon which is in staircase form along both 
axes, 

WETAND DRY MONTHS. Summary of the number of times each month has been first, second, 


third, etc., in order of humidity during the years 1868 to 1906, 38 years inall. Taken from flow of 
Croton River, N. Y., at dam. 


Fig. 489. Staircased Frequency Surface. 


To make a physical model of these two-dimensional curves 
is not difficult, but is rather tedious. Different methods have 
been recommended when the solid model is desired. A simple 
practice is to cut strips of wire and mount them vertically 
upon a board, the lengths of the wire emerging out of the board 
representing the z-ordinates, each wire being cut off at the 
point where it would intersect the curved plane or the indi- 
vidual curve. If the board has been previously drilled with 
holes to admit the wires, the holes being at the intersections 
of the two sets of abscissae, it is not difficult to erect in a very 
short time a forest of these wires, their top ends readily out- 
lining the contour and shape of the curved plane. The next 
step is to place the board with its wires in an enclosed box and 
pour enough plaster of Paris over it to cover all the wires. 
The last step is to cut away the plaster of Paris until the ends 
of the wires appear, the plaster of Paris being easily scraped off 
so as to form a smooth plane. It is also a convenience to out- 


FREQUENCY SURFACES 655 


line the various horizontal co-ordinates upon the sides of this 
solid model and upon the curved plane at the top of the model, 
where these horizontal co-ordinates intersect the curved plane. 
These co-ordinates can be shown by thin black lines thus 
facilitating interpolation and the reading of plotted values 
from the model. 

Another method by which the same kind of solid model 
can be obtained 1s to plot the individual curve upon pieces of 
stiff paper, one complete set of the curves being plotted so 
that they can be set up side by side in the same way that the 
strings were attached to form vertical planes in the network 
in our imaginary room. When the variations of these different 
curves are great, it is also of advantage to plot another com- 
plete set of curves of the same data upon the other independent 
variable, this second set of curves showing the appearance of 
the connecting strings finally added in the imaginary room 
above discussed. In this case, we have two complete sets of 
curves, one for the longitudinal curves and the other for the 
latitudinal curves. By cutting slits in the heavy paper as 
described in the last chapter, the two sets can be fitted to- 
gether as the divisions in the EE UIE egg box in which you 
buy a dozen eggs. 

If the paper upon which the curves are plotted is cut away 
at the curve, that is, if you take a pair of scissors and cut 
through each plotted curve, the lower halves of the chart will 
have the outlines of the curve for their top edges and when 
they are joined together like an egg box, they will form a three- 
dimension model which can be collapsed, if a collapsible model 
is desired. If a rigid solid model is desired, you can pour 
plaster of Paris into this collapsible model, scraping the surface 
_ of the plaster of Paris down to the edges of the paper curves so 
as to obtain the smoothed curve plane. 

+ When the curved plane is to be in staircase form, it is easily 
built up out of blocks of wood. ‘The procedure here is very 
simple. You need merely take a long piece of finished lumber 
with a square cross section and cut it into strips the same length 
as the wires which you would have left standing in making a 
plaster of Paris model. The wooden rectangular cubes are 
then glued together to form a single solid model. There is no 
need of projecting the ordinates upon this model, as the edges 
of the individual pieces of wood indicate the ordinates, but it 
is well to mark inthe horizontal rulings around the side of the 


656 CHARTS AND GRAPHS 


model so that the height of the individual pieces of wood can 
be easily estimated from an examination of the model. Need- 
less to say, in all types of solid models the scale calibration for 


From R. E. Scott, in Harvard Engineering Journal, by permission, 


Fig. 490. A Solid Model—Rounded. 


Model showing cost of light in cents per 1000 Candlehours with 40-watt “Mazda” 
lamps, for any combination of efficiency and smashing point, where price of 
lamp is 50 cents and of current 10 cents per Kilowatt hour. 


the horizontal distance should be around the edges of the bases 
of the models. 

Both the smooth and the staircase curved planes can be 
often easily pictured upon isometric or other axonometric 
paper as described in the previous chapter, eliminating the 
cumbersome and tedious y constructed solid model. In 
general the staircase form of curve-plane is perhaps more 
easily projected upon this paper than the smoothed surface. 
The paper has the disadvantage of presenting the view of the 


FREQUENCY. SURFACES 657 


model from one side only. Therefore care should be used in 
the selection of the side from which the model will be seen on 
isometric paper, in order to get as much detail as possible, 
that is as many parts of the curved plane visible as possible 
upon the isometric drawing. The isometric drawing is per- 
haps best adapted to symmetrical forms of double frequency 
series or to the double ogive or cumulated double-frequency 
series, for in the case of ogives the variation can all be seen 
from one side anyway. A third dimension can of course be 
symbolized upon the ordinary two-dimensional chart by the 
use of colors or shading which epitomize the staircase form of 


Bebe) 
het} ee 


468 Q8OR 


~- 
e 
e 
r 


lan 6 Feb r Qe J Jul A Se Oct Now Des 
J Me Ap’ Mey reer ug Pp 


WET AND DRY MONTHS 
Summary of the number of times each month has been first, second, 
third, etc., in order of humidity during the years 1868 to 1905 
38 years in all. Taken from flow of Croton River, NY., at date 


Fig. 491. An Orthographic Model. 


curved plane, or by orthographic lines (similar to contour lines 
in topography, or to isothermal lines in weather maps) which 
zone off the smoothed curve plane.1 

In general the frequency surface has for its two horizontal 
dimensions, that is, the axes of its two independent variables, 
a rectilinear pattern of co-ordinates. It may be, however, 
that upon these co-ordinates a series of irregular shaped out- 


1 See Chapter LIV. 


a=." 


a 
: 


e 


CHARTS AND GRAPHS 


lines are traced, which are the boundaries of irregularly shaped — 
areas to which our dependent data (the function or body of 
the tabulated double-frequency distribution) attaches. The — 
map is a special case of this in which the horizontal co-ordinates 
mark off longitudes and latitudes and the areas represent — 
geographical localities. The orthographic chart is a general 
case of this in which the irregular outlines are called contour 
lines and the areas are merely zones of equal functional value. 
On the other hand, it is not necessary that the horizontal co- 
ordinates be rectilinear to begin with; thus we may use tri- 
linear co-ordinates, such as are used in the hundred per cent — 
triangle, for our horizontal plane or base. Indeed when the 
100% triangle is used we have a peculiar chart showing four 


From John B. Peddle’s ‘Construction of Graphical Charts,” by permission. 


Fig. 492. A 100% Triangle Model—Four Variables. 


Professor Thurston’s solid tri-axial model showing the efficiency (by height) of 
alloys of three metals in various proportions. 


variables, three of which combine by addition (or logarith- 
mically, by multiplication) to form a constant.2 The solid 
built up from a 100% triangle, might by its altitudes (or 
z-ordinates) show, for example, the efficiency of foods whose 
composition is indicated, as to fats, proteids, and carbo- 
hydrates, for example, by position horizontally. Various pro- 


> Cf. Robert Thurston, Glyptic Models, in the Transactions of the American Socic.y 
of Mechanical Engineers, 1898. 


FREQUENCY SURFACES 659 


jections of the horizontal scales (for the independent variables) 
are possible, including even a probabilities or double-probabil- 
ities projection. These obviously have but limited usefulness, 
in rore complex cases of equating the phenomena with these 
variables, the probabilities projections being designed to con- 
vert the ogive surfaces into flat tilted planes, and the other 
projections having the same objects for other series, all with a 
view to the writing of equations. __ 

It is difficult to adhere strictly to the field of chart making 
in these more interesting and important mathematical charts. 
We are constantly tempted to wander off into the field of 
statistical methods with which these charts are sometimes 
intimately connected. And though this book is not a manual 
of statistical methods we shall here digress long enough to 
sketch in a few of the more important uses of the double 
frequency curve. For in the statistical laboratory the fre- 
quency surface is often used for the study or presentation of 
correlation and association between different methods of classi- 
fication of the same phenomena. Correlation between two 
historical series can, as you have seen, be best shown by the 
juxtaposition of their rate-of-change (i.e. logarithmic) curves. 
But correlation between two independent variables of the 
same data can be shown in detail by the stereographic or 
three-dimensional model. 

The nature of the normal curve of error, that is, the dis- 
persion about a central or most typical point, which is to be 
expected under the operation of the law of chance variations, 
has already been explained in so far as variation along one 
dimension is concerned. But variation can equally well take 
place along both dimensions when the nature of the phen- 
omenon is such as to permit it. Thus the dispersion of gunfire 
from cannon varies as to both distance and direction (range 
and deflection). The coaction of the same probable dispersion 
along both axes, that is both longitudinally and latitudinally, 
results in a cone-like peak whose sides appear to slope along 
the curve of normal error, when seen from any side. 

When the double-frequency curved plane presents this form, 
the thought is naturally suggested that the two independent 
variables upon which the grouping or classification of the 
functions depends, are really independent in their action, and 
do not affect each other. When, instead of a cone with its sides 
following the normal curve of error, we have a ridge diagonally 


660 CHARTS AND GRAPHS 


across the chart, whose cross section may or may not resemble 
the curve of error, we have a rough means of measuring the 
correlation between the two bases, the very narrow ridge pre- 
senting high correlation and the wide-spread irregular ridge 
showing low correlation. 


From G. U. Yule, ‘Theory of Statistics," fourth edition, published by J. B. Lippincott. 
Fig. 493. The Normal Frequency Surface—Rounded. 


The two-dimensional curve or curved plane, either 
smoothed or stepping, is perhaps less often used in business 
than it ought to be. It gives to important data a valuable 


projection, changing through time and conditions in the case 


of historical frequency series, and illustrates the co-action of 
the two independent classifications or changing conditions in 


the case of the double-frequency series. The labor of prepar- | 


ing the charts is not great and the illuminating pictures they 
present are ample recompense. 


; 
| 
. 
q 
: 


CuaptTer LVI 
RELIEF MAPS 


It is in the more elaborate form of maps that we find the 
most frequent and perhaps the most generally understood 
form of three-dimensional charts. Every one is familiar with 
the relief-map model used in the school room, in which the two 
horizontal dimensions are used for the latitude and longitude 
as in the ordinary map, but in which the varying heights or 
altitudes of the model indicate the altitude of the land, 
mountains being shown by ridges, and rivers and valleys by 
cuts and hollows. 

Business men and economists have perhaps little interest 
in the physical contour of a country, but the principles of the 
relief-map can be used to illustrate a large variety of other 
things than the actual height of the land levels. The sales 
manager may be interested in a relief-map in which the z- 
ordinates, that is, the vertical distances or heights of the relief- 
map model indicate the density of sales, as shown by per capita 
sales, per dealer sales, or by other means. The engineer may 
be interested in a relief-map in which the height indicates the 
amount of natural resources, water power, mineral deposits, 
and so on in the various localities. “The economist may be 
interested in a relief-map showing the financial resources, 
wealth, crop yields, or other sociological conditions in the 
locality. 

The usual way of presenting these relief-maps upon flat 
surfaces is to indicate the height which the actual relief model 
would have in its various portions, by different kinds of 
colors or shading. If colors are used, they should be arranged 
along a color chromatic scale so that the colors themselves, 
by their changes (for example, from red to blue, through 
orange, yellow, yellow-green and blue-green), have a natural 
significance and can easily be understood. If the colors are 
carefully chosen for their tints and intensities, they can be 

661 


662 CHARTS AND GRAPHS 


successfully photographed, and will show on the photograph 


as black for the reds and white for the blues, and varying — 


through dark grays to light grays for the intermediate colors. 

The problem here is to secure color tints which have not 
increasing optical intensity but increasing actinic intensity, 
for the camera does not photograph different colors precisely 
as the eye registers them. Many attempts have been made to 
adopt scales of increasing intensity of color, regardless of their 
chromatic sequence, but these attempts are almost always un- 
successful because of the extreme variation of available colors 
which may be used with apparently the same optical results, 
but with actually different actinic or photographic values. 
Moreover, the arrangement of colors solely according to ocular 
density or intensity is unsatisfactory because for the signific- 
ance of each color the reader of the chart must refer to a key. 

There are many different ways of applying color to charts. 
The disadvantage of water colors is that they tend to run 
upon the paper and are difficult to shade off, no two mixtures 
being precisely alike when laid at different times on different 
charts. Colored water-proof inks, ordinarily used in drafting, 
must often be diluted or their intensity will be so great as to 
hide any printing or labelling which was intended to show 
through the color on the chart. For extreme transparency, 
photographers’ Japanese transparent inks and lantern-slide 
colors can be used. Perhaps the best results are obtained in 
general with ordinary wax crayons, the cheaper and more waxy 
they are, the better. These can be laid on very thickly and 
evenly, and can then be scraped away with a sharp knife 
edge, leaving a delicate tint through which all printing and 
labelling will be easily seen, and not warping or wrinkling the 
paper. 

Shadings are of many different kinds. The Census Bureau 
makes great use of dot shading, a number of small dots being 
placed upon the paper, scattered over the locality and by their 
number, showing the values of the data for the locality. The 
method has a decided advantage in that the charts tend by 
their crowding or scattering to indicate density visually, but 
it 1s a tedious method to follow in the making of the chart and 
the results do not afford any degree of accurate reading. It is 
impossible for the average reader to count the number of dots 
where these have been thickly placed and where they are so 
thick as to form almost black areas the significance of the 


—— 
i 
| 


dot has become entirely pictorial. The method is, however, 
far petter than another one which resorts to dots or circles of 
various sizes in which the areas inscribed in the circles indicate 
the values of the data. For in these circles of various sizes, 
we meet with optical illusions and difficulties of accurate chart- 
reading as well as chart making, described very early in this 
book. The area of the circle although a two-dimensional 
measure, is used to illustrate one-dimensional data. Dot- 
maps are often most easily made with colored map pins or 
map tacks which have been described in an earlier chapter. 

Much the better form of shading is secured by a careful 
scheme of hatching and cross-hatching lines so arranged as to 
give an optical effect of shading from white to black and at 
the same time sufficiently different in pattern to be easily 
identified from a key or appended scale of shades. 

Several such patterns of useful shadings have been designed. 
The work of drawing in these shadings, however, is sometimes 
very great, it is difficult to rule these hatching lines uniformly 
without’ a special instrument (section-ruler) and the whole 
process takes a great deal of time. It is therefore sometimes 
better for the average chart which must be prepared in black 
and white, to mix India ink and Chinese white in various 
degrees and ink them onto the chart as so many tints of gray. 
When this is reproduced photographically the effects are en- 
tirely satisfactory for photostats, but are useless for blue- 
prints. As these tints require half-tone engravings for printed 
reproduction, it is more convenient, When the map is intended 
solely for printed reproduction, to use the forms of cross- 
hatching and shadings known as “Ben Day” in the printing 
office. When Ben Day is used, your original drawing need 
have no shading at all, the various types of Ben Day merely 
being indicated by numbers or symbols in blue pencil on your 
drawing; the engraver will insert them properly. 

Relief-map models may be compared to the smoothed- 
plane curve of the last chapter, in that the changes of types 
are never abrupt but are gradual. The staircase form of these 
map models is an elevation map or table-land map which is 
less often seen. It can be best constructed with a few sets of 
maps, mounted upon boards of different thicknesses, and cut 
or sawed apart along the State boundaries. Children’s puzzle 
toys are sometimes made in the form of maps of the United 
States in which the individual States have been cut away in 


RELIEP MAPS 663 


664 CHARTS AND GRAPHS 


this fashion, and may prove useful for this purpose. A picture 
of the elevation or table-land map is, however, easily drawn 
upon isometric paper, or upon plain paper, by tracing the 
State outlines from a regular map of the United States, after | 

{ 


shifting the position of the tracing paper slightly to correspond 
to the representation of height or elevation of each State. 
Such maps are very effective in their way. 

One disadvantage of the colored or cross-hatch map upon | 
paper drawings is that it represents a staircase form of map . 
while most phenomena should really be smooth, as the transi- , 
tions from State to State are not abrupt but gradual. To meet 
this problem, the colored or shaded map can often be skilfully 
converted into a zone map, in which the colors or shadings 
have been zoned so that no two colors or shadings appear side 
by side upon the map except those which are consecutive in 
the key or scale of colors or shading. Where the data applies 
to the entire State or other territory, these zonings are of course 
arbitrary and tend to alter significant areas in precisely the 
same way as the smoothing of a frequency polygon destroys 
significant partial areas of the polygon. The zoning should 
therefore be made very narrow, along the outlines of the 
States, in order to leave as large as possible a portion of the 
space properly colored according to the data. 

'«« However, if the data does not represent the entire State 
or other territory, but merely indicates conditions at certain 
points, such as certain cities, the zoning should always be done 
and the zones should be of equal width between any two 
observed points. A familiar example of this type of zone map 
is to be found in the map used by the Weather Bureau showing 
high and low pressure areas from day to day and temperature 
lines across the country. In fact the entire zoning process is 
merely an attempt to reproduce for the particular data, the 
same excellent results achieved by these isothermal lines upon 
the weather map. The contour lines upon topographical maps 
are examples of the same type of zoning or orthographic rulings. 

The technique of these various presentations of the third 
dimension upon the map is fully described in a previous chapter 
and we have here only hastily recapitulated the more common 
forms of maps. In the case of all the maps so far considered, 
the reader will notice a limitation, in that each map is capable 
of presenting but one set of data. Two figures for each locality 
cannot be shown upon the same map. There are, however, 


| 


? 


RELIEF MAPS 665 


two ways in which more than one geographical distribution 
can be shown upon the same map. The first of these methods 
is the very obvious one of combining on the same map two 
methods of showing the third dimension. Thus on a flat map 
both colors and cross-hatchings can be used simultaneously, 
the colors to show one set of variables; the hatchings, another. 
The results are not wholly satisfactory. If bars or other area 
charts (circles, stars, etc.) are used to show values of course a 
series of bars or areas can be used in each part of the map, 
corresponding bars being perhaps distinguished by color or 
shading. If a stereographic map (i.e. solid model or axono- 
metric drawing) be used, of course one set of values is shown 
by the altitudes (z-axis ordinates) and another by colors or 
shadings drawn upon the resulting surface. And always the 
use of numbers actually entered upon the map may give us 
further values not graphically displayed. All of these methods 
give us what might be called multiple maps, in that they are 
combinations of two or more map surfaces. The second way 
of showing more than one geographical distribution is more 
laborious in construction, but also much more illuminating. 
It is the method of the bead-map. It gives us not only multiple 
maps, that 1s combinations of distinct distributions, but also 
it gives us compound maps, that is segmentations of a single 
distribution. ‘This last feature is one which cannot be satis- 
factorily achieved by any other graphic methods and can be 
shown only on a flat map by figures, The bead-map must be 
classed with solid models, for it is a rigid body occupying space 
in three physical dimensions as fully as if it were of plaster-of- 
paris, wood, or some other substance. In its construction we 
fall back upon the upright wires which were used in the con- 
struction of the plaster-of-paris model. 

The bead-map takes its name from the fact that after 
properly cutting the vertical wires, it is customary and most 
satisfactory to string beads upon them before their ends are 
inserted in the map. Beads can be obtained for use in this 
way, in many different colors, at the average department 
store. Sometimes glass beads of uniform sizes, especially 
advantageous for this work, can be obtained from the publish- 
ers of charting material. The wires themselves should be fine, 
of medium strength and spring and can also be obtained from 
charting-material publishers, especially adapted for this work. 
Where only a few beads are to be strung and the wires do not 


666 CHARTS AND GRAPHS 


extend very far above the paper, very long and thin steel pins 
such as are used in natural history museums, can be used for 
the wires, the heads of the pins holding the beads and prevent- 
ing their escape. When wires are used, small knots must be 
tied at their upper ends, to prevent the beads from coming off 
the wires. The beads on each wire should be all of one color 
except that every tenth bead, or (according to the scale for 
beads), every significant bead, should be of a contrasting color 
so as to facilitate the counting of them by the reader of the | 
map. 

These maps can be made with as many as three or four 
sets of separate wires for each State in the Union, each set 
representing a certain figure or set of data, on maps which 
+hemselves are no larger than ordinary-sized letter paper, that 
8, 8¥ by 11 inches. The different wires for each State should 
be placed upon a single line or row across the State so that they 
can be easily compared and form, as it were, a vertical bar- 
chart upon the State. 

The variety of purposes to which the bead map can be put 
is as great as the uses of other relief or color maps. The sales 
manager, for example, will be interested in a map in which 
there are four columns of beads in each State, a column of 
green beads indicating the population or potential market in 
each State, a column of red beads indicating the sales of his 
competitors in the State, a column of blue beads showing his 
own sales in the State in the previous year, and a column of 
black beads showing his sales this year. A comparison of the 
red and black beads shows him how well he is keeping up with 
his competitors, while a comparison of the green and black 
‘beads shows him how well the market is being saturated by 
his goods, and a comparison of the blue and black beads 
shows him his last annual increase or decrease of sales. 

Another convenient form of this map is sometimes called 
the “tree-map.”’ In this the stringing of beads upon wires 
has been eliminated entirely and small pieces of colored wood 
sticks substituted for them. A full equipment of wood sticks 
of uniform thickness and shape, but of different lengths and 
colors, can be obtained from the manufacturers of kinder- 
garten toys, being often sold for kindergarten work. When 
these sticks are being used, their ends, to be inserted in the 
map, should be sharpened with a knife so that they can be 
easily inserted, after first marking upon each clearly the dis- 


J 
{ 
q 
. 


RELIEF MAPS 667 


tances which should be left exposed, sticking up out of the map. 
After they have been driven into the map to the proper dist- 
ance, they should be removed and into the holes made by 
them drops of glue should be placed, the sticks being then 
replaced in the holes and allowed to set in the glue. The tree- 
map does not afford the possibility of exact reading which 
was possible in the bead map, where the beads themselves on 
any wire could be readily counted. If this feature is desired, 
small colored bands should be drawn at the points about the 
sticks of wood at the heights where the distinctly colored 
beads would appear, marking off on each stick of wood the 
ordinates of the various convenient values on the z-axis. 

Needless to say, the map for beads and wires, or for sticks 
of wood, should be mounted in the same way as pin maps, 
which have been described in an earlier chapter. They should 
have at least three layers of corrugated pasteboard under them 
to hold and protect the ends of wires or sticks! Neither 
bead-maps nor tree-maps are convenient to file or to have 
about in large numbers as they are apt to get damaged. The 
best way to file them or to carry them about is to use small 
wooden boxes or cases into which they can be slipped easily 
and fit compactly, and in which they are prevented from 
moving about by small retaining flanges inside the boxes or 
cases. 


tda. 4.5 Mant, 220 


Gis oi 
KEES 

Aria BE orsign 
NM. 1966 VCP X KS 


LI 
SOL 
ROK KK ROSCA 
ROG OS 


SCHOOL TRUANCY 

fercentage Ff Children of School Age 

(1-13 yea, incl.) Not Attendims Schoo| 
Un ted Srates (4 Te) 


(seer 12 ana) 
Note: For other Ages the percentages for The Yaitid Hed au: 


VA and 18 years 6) t— 26.1 % 
16 and nn - Sy S71 
1S. te aie |} os.a 


Fig. 494. Cross-hatched Map on the Population Projection. 
ee eee ee 


1 See Chapter IV. 


668 CHARTS AND GRAPHS 


Keys should always be provided with every map, to 
explain the significance of colors, shadings or beads, and wooden 
sticks. These keys serve the same purpose as scales in curve- 
or bar-chatts. They should be complete and carefully worded. 
It goes without saying that the special projection maps de- 
scribed in the chapter on population maps, can be used in the 
place of the ordinary land-area map for all cases where the 
significance of map areas is better shown by such projections. 

Maps need not be used only for the display of character- 
istics of entire localities and territories, but can also be used 
for the analysis of routings and conditions along certain 
routes or at certain points on the map. In this case we are 
not concerned with areas on the map but with lines or points 
upon them. The use of strings upon maps, connecting map 
pins or map tacks, has already been described in an early 
chapter. These strings can be used not only to indicate 
actual routes which will be followed by sales managers, 
travellers, or trafic, but can also be used to indicate spheres 
of influence, authority, or other connecting influences. Thus 
the circulation of a number of newspapers situated at different 
points of the country can be shown by strings (or indeed by 
mere ink lines) radiating from their places of publication to 
the residences of their furthest subscribers, different colored 
strings (or ink) being used for each newspaper. Such a map is 
sometimes useful in the analysis of newspaper circulation for 
advertising campaigns. Likewise the line of authority from 
central office to the various branch houses and from branch 
offices to the individual agencies can be similarly shown by 
radiating strings. Such maps properly belong to the class of 
combinations or superimpositions of route-charts upon maps 
described in the chapter on combinations of non-mathematical 
charts. 

When, however, we attempt to show the volume of traffic 
or travel, or the extent of any other connecting phenomena 
between two points upon the map, we come quickly into the 
field of three-dimension maps. If we wish to present this 
graphically, a very effective method has been found of pre- 
paring ribbons of stiff, colored cardboard or paper, and mount- 
ing these ribbons on edge along the route or line of traffic or 
connection, so that the height to which the ribbons rise, will 
indicate the volume of traffic or other connecting phenomena. 
The same result can be more easily obtained by the use of 


| ie 


RELIEF MAPS 669 


colored strings connecting columns of beads which have been 
previously erected at the points to be connected. ‘The strings 


a 
Tr onens 


y 


any 
mera 


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4 
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) 
\ 
my 


A 
APACHE : 
Pe 2 
ta 
Fes elem 
Sp Nl ST | 
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From “Electric Railway Journal,”’ and Haskell's ‘‘How to Make and Use Grapizic Charts,”’ by permission. 


Ou = 
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By 
3 


(ees | 


BAY 
Ag 


are easily tied to the columns of beads and run back and forth, 
one string between each layer of beads, forming fences similar 
to the old-fashioned rail fences and indicating by the number 
of strings or rails the volume of traffic or other figures for the 
connecting phenomena. If we wish to present the three- 


670 CHARTS AND GRAPHS 


dimensional data upon a flat surface, for convenience in 
handling and filing, and the number of routes or connection 
lines is not great, we can show the comparative height of the 
various ribbons or set of strings by colored or shaded bands 
drawn upon the map connecting the points which the bead or 
string fences would connect. In this case, the widths of the 
bands representing the volume of traffic or other connecting 
phenomena. 

The reader will have seen by this time that map-charts 
are almost a field of charting in themselves, with wide diversity 
and flexibility and an infinite variety of forms, capable of the 
widest variation and adaptation for special purposes. In fact, 
the map can be considered as distinct from all other types of 
charts in that the fundamental two dimensions, that is, the 
two dimensions of the base-map itself, are used solely for the 
purpose of displaying geographic position and location (except 
in population maps) and not for strictly mathematical relations 
and that the mathematical relations must be charted upon 
this ground-map by the use either of a third dimension or of 
superimposed drawings representing a third dimension. In 
a sense therefore, the map can be considered an inefficient or 
wasteful type of chart. For economy of space and charting 
dimensions, the superior form of chart for all data having a 
geographical basis is the bar-chart in which the geographical 
location is shown by a list of stubs and the independent 
variable, that is, the geographical location, occupies only one 
dimension on the paper. This applies not only to maps, but 
also to diagrams, floor plans and other illustrations of space 
or physical localities, all of which can be treated in the same 
way as maps have been treated in this chapter. But in spite 
of these disadvantages and inconveniences, the map is so useful 
that it can be strongly recommended to the studious chart- 
maker as a powerful method of displaying such facts as are of 
sufficient importance to justify the greater labor and care 
involved. 


ening 2 


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Cuaprter LVII 
THE STATISTICAL MATERIALS 


Few, perhaps, of our readers, have run the gamut of chart 
forms and methods which we have described in this book 
without realizing that there is almost as surely a natural 
evolution in charts as there is in other sciences or arts. It is 
possible and would indeed be interesting to construct a dia- 
gram in the form of a tree-chart, showing the development of 
each chart form out of common root-forms. Within the 
bounds of our limited ability we have constructed this book 
after such a pattern. And as time passes and new forms are 
invented, or new modifications are introduced, it will always, 
doubtless, be possible to relate them to existing forms and allo- 
cate each to its proper niche upon such a diagram. 

But more interesting than the classification of chart- 
forms is the classification of the statistical materials which 
they illustrate. For the numerical arrays and tabulations, 
the counts, samplings, enumerations, ‘and reports which we 
call statistics present even greater variety and heterogeneity 
than the charts by which we may picture them. Nor need 
such a classification be wholly academic. The coding and 
systematizing of graphic methods can hardly progress far with- 
out becoming tangled in the chaos of statistical forms, a con- 
fusion from which it cannot again escape until we have set to 
order the statistical stock-room. 

If then, we could succeed in so neatly classifying and 
pigeon-holing each type, species and hybrid form of statistics, 
that the novice could readily identify each specimen, we would 
set for ourselves this aim: That each variety of “‘statistic’’ 
should be clearly labelled and marked with the one, two, or 
more ways of charting suitable for its illustration. We would 
have this code, key, or system, so simply set forth that the 
economist and the business man, be he ever so untutored in 
the science, could easily locate in it his particular bit of statis- 

671 


672 CHARTS AND GRAPHS 


tics and as quickly set out effectively to chart it. This we say © 
would be our ambition, were it possible. And while many | 


may doubt its possibility, yet in this chapter we shall venture 
a few first steps in its direction, only bespeaking i in our readers 
a tempering of judgment with generosity for the short-comings 
and failures which attend us. 

We do not progress far in statistics without noticing that 
all numbers are purely adjectival, and that to each number, 
in order that it may have a meaning, a substantive must be 
attached. If we only make mention of so small a thing as 
two pins we may observe that the numerical adjective “two” 
holds meaning only when attached to the noun “pins.” Speak 
of three needles and we note that in this beginning of a statis- 
tical collection we have changed both adjective and noun. 
To make it look quite professional and uninteresting we should 
tabulate it, thus: 

Vinge... tae cee ee 

NC eee 
But suppose that these pins” were ordinary household pins, 
while we may discover elsewhere five safety-pins, to be added 
to our collection of pins, bringing it up to seven, thus: 


Pins, common..... Zz 
Pins, uk AN 
Needles. . Peers 


Now we notice that while thee noun has remained the same, 
another adjective has been added beside the numbers. Many 
such qualifying additions could be made. And it is the object 
of this homely illustration to point out that while numbers 
are always adjectival, not all adjectives are numerical, and 
that numbers like other adjectives, such as “safety” or 
“common,” have a meaning only when associated with some 
substantive. 

But we stand too long on pins and needles. Let us only 
keep in mind the point that in all statistical work we must 
early recognize two variables, variates, or variable facts, 
which for convenience we have always distinguished as inde- 
pendent and dependent. It may be that at times one will 
seem the noun and the other the adjective, or that various 
adjectives will vie with each other for importance, and that 
at other times in the same data a reverse arrangement will be 
more useful to us, but always at one particular time we must 
use one variable as independent, that the other, clinging to it, 


' 
i 


—— es 


PoE STATISTICAL MATERIALS 673 


-may ve dependent. And statistical technique is largely a 
matter of the proper marshalling, commandeering and buffet- 
ing about of these two sets of variables until they behave in a 
way that yields up to us an intelligible message. There is no 
hard and fast distinction between the two variables and their 
degrees of independence or dependence—they are truly inter- 
-dependent—but always there are hard and fast rules which 
govern the treatment of them in such a way, that whichever 
variable is playing the independent role may be given such 
and such handling, and the other which is playing the depend- 
ent role has such and such other possible operations. As a 
rule we usually let the independent variable take its own 
course and put most of our efforts upon the dependent one, 
but this is not always so. 

Another thing which we may note at once is that statistics 
may come to us either in singular or plural form. We may 
have, as it were, a single “statistic,” or a collection of statistics. 
Thus the simple fact: 

Pitise ns Ore et 7 
may be the alpha and omega of our desired information; or the 
more elaborate statement: 

Pins, conimion’. 2 ..% 2 

Pins, safety 5 
may be our objective. Here we come upon the distinction 
between what may in a wider sense be termed ‘‘averages”’ and 
“‘distributions.”” Commonly, this particular average would be 
called a “total,” for it is the total of its parts, which parts are 
forcibly brought to mind by the subsequent distribution. And 
in a specialized sense, averages and totals are very distinct. 
Thus we would say: 


Pins, total. [2200s 7 
Pins, common! 2 
Pins,vsatety *. 2985 
Pins, average......314 


But in a wider, perhaps more precise sense, all so-called totals 
are merely averages—if you will, averages of the counts taken 
of the items, or averages for particular selections or samples. 
Into the various kinds of averages we shall not delve—their 
consideration forms the large part of most elementary treatises 
on the science of statistical methods. Let us compromise by 
distinguishing at once single and collective statistics as “‘aver- 
ages or totals’ and “distributions.” 


674 CHARTS AND GRAPHS 


Now before going on to collective statistics, or distribu-— 

tions, let us look just a little longer at the single entry or item: 
Pings:ii iv saley leew 
We note that the number, 7, is a count. But a count of what? 
A count of the number, you say, of whole pins. So itis. And 
being so, it is so obvious that it has not been included in the 
statement. Could we count these pins in any other way? 
Suppose we count the half-pins, then there are fourteen in all. 
But this is quibbling, you say. Very well, suppose I inform 
you that these pins are exceptionally rare and are valued at 
a dollar apiece. Now you can count them as: 
RINE Molen poatleen ee 
or more fully: 
Pins (value) 7 (dollars) 
Again we discover that they are railroad coupling pins and 
weigh two pounds each. Now we can write: 
Pins (weight) 14 (pounds) 

And all the time we are merely describing in different ways 
the same objects! In short, in statistical work, we deal with 
various numbers and by them we count various items in terms 
of or by means of various units of measurement. The units 
may be, roughly, measures of volume or of value. Volume 
statistics refer to physical volume in a loose sense and may be 
in terms of linear, surface, content, weight, or other measures, 
including the mere count of the number of items. Value 
statistics refer to intrinsic worth as indicated usually by actual 
or potential price or cost, and are generally in terms of money 
of one currency or another. 

While the separate or isolated average or total may appear 
in any of these forms, it does not in any of them become a 
proper subject for a chart, for a chart is only useful as a means 
of comparing two or more figures. We do not enter the field 
of charting therefore, until we come to collective statistics or 
distributions. In turning to these, however, we must keep in 
mind the various possible forms of the individual item or 
“statistic’”’ for it is obvious that they apply as well to the collec- 
tion. The collection or table of items, then, will contain both 
independent and dependent variables, and will be composed 
of either averages or totals (now become sub-totals), measured 
in units of volume or value. In fact, the collective statistical 
statement often appears to be no more than a mere agglomera- 
tion of individual statements placed together upon a page. 


THE STATISTICAL_MATERIALS 675 


_ At other times the collective statement appears to be a more 
detailed elaboration of some individual statement. In the 
latter case the name distribution is clearly called for, but it is 
also true that the most patently heterogeneous aggregation or 
conglomeration of figures can generally be regarded as a de- 
tailed distribution of something or other. In this sense we may 
_ speak of the simple individual item as “undistributed” and the 
collection of items as ‘‘distributed.” 

It is in studying the various types of distributions that we 
strike the first important distinctions of data, and, by corollary, 
of charts. Four types at once come to mind, which, for con- 
venience, we may call respectively, abstract, geographical, 
frequency and historical distributions. Indeed it is not im- 
probable that in the course of time the compilers of statistical 
volumes will adopt distinct tabular forms for each of these 
types and consistently maintain these forms in their compila- 
tions. The Bureau of the Census has already adopted a more 
or less standard form of table for geographical distributions 
where these cover the States of the United States. Even 
more successful is the excellent standardization of historical 
tables by the same bureau for its ‘Survey of Current Business.” 
By the side of the usual confusion of tables in most statistical 
compilations the simplicity and clarity of these forms is indeed 
refreshing. 

There is very little overlapping of the four types of distribu- 
tions. The first two are essentially logical, the last two are 
essentially numerical, in respect to thé bases of their distribu- 
tion or classification, which form the independent variables or 
“stubs,” in them. The first and third are generalized types, the 
second and fourth are merely extremely common and important 
species of the first and third. The basis of the second, the 
geographic distribution, is space, while the first, the abstract 
distribution, can have as a basis, any other set of logical rela- 
tions. The basis of the fourth or historical series is time, 
while the third or frequency series can have as a basis any 
other set of numerical relations. And, of course, we can have 
what might be called composite statistical tables presenting 
two or more of these distributions simultaneously. Indeed, in 
most statistical compilations the composite distribution 1s 
chiefly used for its convenience of comparisons and economy 
of space. Not only can two abstract distributions be made to 
interlock in a single composite one, but two geographic and 


676 CHARTS AND GRAPHS 


two frequency distributions are often so combined. And 
there can be any combination of different types. In the present 
book the distribution placed at the sides of the table has been 
called the series of stubs, generally considered the more im- 
portant, and the other, placed across the top of the table, 
has been called the series of column-headings or captions. 

If we should attempt the explicit description of statistical 
distributions of these various types—confining ourselves to 
single distributions only, since what applies to these applies 
likewise to each of them in composite distributions—we would 
find certain salient points which must be noted about the 
independent variables in each type. Thus in describing (or 
cataloging) an abstract distribution the important things to 
note are the basis of the distribution (whether it is nature of 
diseases, causes of accidents, kinds of articles, races of the 
population, sex, marital condition, or what not) and the number 
of items in the series. Little more can be done to categorize 
the abstract distribution. And we may note that in graphics, 
the abstract distribution is amenable to no connected method 
of illustration, such as maps or curves, but is limited to bar- 
charts (including 100% bars and circles, or pie diagrams) and 
area bars. 

In the geographic distribution there are more salient points 
to be noted. First, we should note the whole (such as world, 
continent, country, state, etc.) and the parts (continents, 
countries, state-groups, states, counties, cities, etc.), into 
which the whole is divided or distributed. Moreover, for 
convenience we should also note the number of parts, and their 
completeness (that is, whether or not their sum forms the 
total). Population and sales statistics are often complete, 
building statistics, and morbidity and mortality reports are 
examples of commonly incomplete data (compiled from only a 
few states or cities). Inthe graphic presentation of geographic 
distributions we can use maps in addition to the bar-charts 
which are applicable to all types of statistics, but we cannot 
use curves. For complete data we can generally shade or color 
whole areas on the map, but for incomplete data, such as re- 
ports for various cities, we should use isolated points on the 
map. 

In the frequency series we need not only to know the basis 
of the numerical relations of the series and the number of items 
in the series, but we need also to know something very much 


" ay 


THE STATISTICAL MATERIALS 677 


akin to completeness, namely its continuity. By continuity 
is meant whether the independent variable be a discrete or a 
continuous series, and if continuous, whether it be “point” or 
“period” data. By point data we mean data for separated 
non-contiguous points in the range; by period data we mean 
data for connected contiguous or overlapping periods in the 
range. For all these forms we can use curves, as you know, 
in addition to the ubiquitous bar-charts. But for the discrete 
series, generally composed of integral varieties, we are limited 
to the staircase or rectilinear curve so closely akin to bar- 
charts. Indeed this type of distribution has often little more 
than the accident of numerical designations to distinguish it 
from the abstract distribution. For the continuous series, 
generally of graduated variates, we should give a truer picture 
by using the smoothed curve or frequency polygon, though 
for period data we make a sacrifice of the accuracy of area 
representations thereby. And we may note a further distinc- 
tion that while for discrete and period-data continuous series 
we should plot the data in the spaces between ordinates, for 
point-data continuous series we should plot the data on the 
ordinates. These distinctions have been discussed in the 
chapter on amount-of-change frequency curves. 

Lastly we come to the historical series. Here four salient 
features of the independent variable should be noted. There 
is the range of time (the whole) covered, the intervals of time 
(or parts) used, and hence the number of items, and finally 
their regularity and continuity. The range and intervals are 
in centuries, decades, years, months, weeks, days, hours, and 
so forth. And some tables use different intervals in the same 
distribution, such as a series of decades followed by the indi- 
vidual recent years and lastly the most recent months, making 
in all for irregular intervals. ‘The continuity of the data here 
means simply a point and period distinction. Items for 
isolated points of time, such as stock or balance reports at the 
first of each month, or price quotations at the end of each 
week, are point to point data. These the economists call 
stock or fund figures. Totals for periods of time, such as pro- 
duction or shipment statistics, are period data. These the 
economists call stream or flow figures. And it is of course 
possible to have isolated periods reported without the inter- 
vening periods. All historical data can be presented on curves, 
as well as bars, but in the summary chart the period data is 


678 CHARTS AND GRAPHS 


shown by bars, the point data by curves. In general, wherever 
we wish such a refinement, we can perhaps more accurately 
show period data by staircase curves or vertical bars, and point 
data by smoothed curves. A far more frequent distinction, 
however, is that while period data can be more accurately 
plotted in spaces between ordinates, point data can only be 
plotted upon the ordinates. These considerations have been 
brought out in various portions of the text. 


To the foregoing discussion of statistics as regards the 
independent variable, we have now to add a brief outline of 
what may happen to the dependent variable. And here we 
can no further escape another general distinction which can 
be made between what might be called primary and secondary 
or derived statistics. The primary statistics consist of totals 
or averages which represent the original observations reported 
in the statistical table. Now these totals or averages, which, 
by the way, form the dependent or adjectival variable in the 
table, can be subjected to statistical treatment and materially 
modified, and the statistics which have suffered such treatment 
may be called secondary or derived data. 


There are two main kinds or processes of statistical treat- 
ment which can take place simultaneously or individually. 
For the sake of simplicity they may be called compilation and 
conversion. Statistical compilation is to some extent operative 
upon both the independent and the dependent variables. 
Statistical conversion is almost entirely limited to the depend- 
ent variable and though it is perhaps considerably the more 
intricate subject, will receive scant attention from us, as it 
does not greatly affect the charting method. 


Statistical compilation begins with material in its crude 
form, which is a mere listing or list. In the case of the logical 
distributions this is also about where it ends. Much rearrange- 
ment is possible, of course, but the abstract and geographical 
distributions always remain nothing more than lists. This, 
perhaps, is why neither can be shown in any graphic form 
where connection-lines represent continuity or sequence, in 
short, in any form of curve. The numerical distributions, 
however, can be so arranged that the items follow each other 
seriatim in a sensible way. In the early chapters of the book 
we have spoken of this as a case where the stubs or independent 
variable facts fall into an order imposed by themselves. 


THE STATISTICAL MATERIALS 679 


Obviously it is nothing more than mere mathematical sequence 
in the stubs, which dictates this order. 

When a numerical distribution has been arranged in this 
orderly way it ceases to be a crude list and becomes a special- 
ized one which is commonly called a “series.” It is now usually 
ready for charting in the form of a curve, for there has appeared 
in the data a thread of connection running through the various 
items, a thread which enables us to connect the items on the 
chart by a line. This curved line can be shown on scales of all 
the various types discussed in the sections on curve-charts, 
from simple arithmetic or amount-of-change scales to logarith- 
mic, rate-of-change, and other projections. It is not to be 
thought that the curve is something radically different from 
the bar-chart; indeed, as you know, the amount-of-change 
curve is simply a convenient sort of short-hand symbol of a 
series of bars, and the rate-of-change curves are merely special 
warpings and distortions of the amount-of-change curve with 
intent to bring out hidden relations and features in the curves. 
But it remains true that in the curve chart we are really for 
the first time, able to shift the focus of our attention from the 
comparisons or changes between individual items in a distri- 
bution, to something more complicated, the comparison or 
changes between these changes in different parts of the same 
distribution or in the same parts of different distributions. 
And to make the curve, we must first compile the numerical 
list into a numerical series. 

This first step in statistical compilation is important, 
because to the casual reader it is hardly apparent. Indeed, it 
may be that the layman, glancing over a volume of the census, 
or over any other statistical series, is quite often under the 
innocent impression that the figures he sees just grew, some- 
what as did Topsy. He does not suspect that many days or 
months of study may have gone into the determination of the 
proper group or interval limits in the series, and that over a 
year thereafter whole batteries of clerks and computing 
machines have sorted and enumerated the items in accordance 
thereto. Not all compilations are the result of such great 
attention. Regrettable it is to say, that publications still 
occur in which the raw material, the crude list, is given; but 
the compilation of data into series has not been carried far 
enough. Such lists and imperfect series tabulations are very 
likely to puzzle the student unless he detects the unfinished 


680 CHARTS AND GRAPHS 


treatment and completes the process of orderly series arrange- 
ment. 

The simple series is the major step in the tabulation of 
numerical data but it is by no means the last, if the data be 
of the “period” type. If the data refer to scattered, isolated 
points of time in a historical series or points in a continuous 
frequency series, it may not be feasible to subject it to the 
processes about to be described. But period data (including 
discrete frequency series) in which the periods covered by the 
items are co-terminous and contiguous, can be subjected to 
the familiar process of cumulation and moving total (and 
average) calculation. In effect these processes change the 
separate groups or intervals into overlapping groups. In the 
cumulation, the overlapping is in one direction only; in the 
moving total (or average) (taken in its proper position as at 
the middle of its period) the overlapping is in both directions. 


Slight differences may be observed in the susceptibility of 
the two numerical distributions to these processes of over- 
lapping. Thus the frequency series can be cumulated in either 
direction, either backward or forward, yielding a “‘more than”’ 
or a “less than” cumulative. It may then be plotted in the 
familiar form of the ogive curve, of which both axes may be 
either arithmetically or logarithmically projected and for 
which the probabilities curve is the great analytical medium. 
The historical series, on the other hand, is sensibly cumulated 
only in one direction, and the curve of the cumulative is most 
especially used in the Zee-chart and its bars in the Gantt 
Progress chart. The historical series, moreover, can be sub- 
jected to moving total and average calculations, for any length 
of time or periodicity, and the moving series is to be used in 
curve charts of all kinds, while the frequency series is never 
subjected to this process except in some statistically technical 
calculations of the mode and smoothing processes. The subject 
of moving totals and averages and cumulations has been dis- 
cussed in detail in the text.1 

We come, lastly, to the other form of statistical treatment 
by which secondary data can be derived from primary stat- 
istical sources. It relates normally to the dependent variable 
and is the process of conversion of absolute data into relative 


1 Beyond the cumulative and moving or progressive totals, which are somewhat 
in the nature of integrals within limits, the processes of differentiation and integration 
are not included in this discussion. 


‘ 


THE STATISTICAL MATERIALS 681 


data of various kinds. The absolute data is the original data 
itself, whether statistically compiled or not. It occurs in the 
form of totals or averages, which measure items in terms of 
various units of volume or value. The relative data is always 
the result of comparing this absolute data with other absolute 
data. The latter, with which comparison is made, may in 
general be called the base of the relative data. The relative 
data itself is of several varieties, which we will briefly mention, 
and occurs under many different names. Its relativity is 
usually obvious enough but is sometimes so completely dis- 
guised by ambiguous titles or nomenclature as to be difficult 
of detection and when this occurs its analysis may prove a 
baffling problem to the inexperienced. 


The most familiar form of relative data is the percentage, 
or series of percentages, of which the total is 100%. It is the 
result of comparing the parts to the whole, or the items in a 
list or table to the sum thereof. In particular we may note 
that the base is common and constant, the same base being 
used for each and every percentage. Hence the percentages 
bear the same relation to each other as the original numbers 
or quantities which they represent bear to each other. The 
percentages, in fact, are but the same statistical facts reported 
in terms of a new unit of measurement. And so the percentage 
series may be subjected to cumulative and moving total (and 
average) calculation almost as readily as the original numerical 
quantities. ' 


The next important type of relative data is that to which 
the special name “relative figures” is given. These share with 
the percentage figures the feature of a common and constant 
base; but differ from them in the relation to their base. 
Relative figures are not to their base-figure as parts to a whole; 
their sum does not total one hundred per cent. The relation of 
relative figures to their base is an item to item relation; that 
is, it is the result of comparing the original numbers, sometimes 
called by distinction, the “numerical data,” with one of the 
component items. When the base-figure is more or less ima- 
ginary, being a combination of several often incommensurable 
base-figures, the relative figures are called “index numbers, 
though the latter term is by some writers loosely applied to all 
relative figures. Owing to the use of common and constant 
bases (for each series), relative figures and index numbers can 


‘ 


682 CHARTS AND GRAPHS 


be subjected to moving total and average calculation, but 
their cumulation is usually of no value and significance.? 

The last group of relative data differs from the foregoing 
in the use of various and different bases within each series 
instead of acommon and constant base. Such series are always 
formed by comparing the items in one series with corresponding 
items in another series. Hence the items in the second series 
are the bases for the items in the resulting relative data. 
Where the latter stand in the relation to their bases of parts 
to wholes, they form percentages. Where they are not in this 
relation, the most common result is a per capita figure, “per 
family,” ‘‘per dealer,” etc. In either case when the fraction of 
ratio is very small it is usual to multiply by a thousand or some 
other constant, and so achieve a “rate.” These relatives come 
in a wide variety of ways and under an equally wide variety 
of names. But they all have in common the feature of shifting 
inconstant bases. And as a result they cannot ordinarily be 
cumulated or smoothed by the moving total (or average) 
process. When we desire to compile them in these ways it is 
only proper to return to the original data and the base-figures 
and perform the operations upon them, that the smoothed 
relatives may be secured from their comparison. 

At this point we draw to a close a very rapid survey of 

types and varieties of statistics as such. It is our hope that 
the reader will find such a panoramic view of value in his 
statistical work and the charting problems that arise therefrom. 
It is our hope that in time the classification and cataloging of 
statistical forms will become so simplified and improved that 
it can be used with immediate profit by the novice, as floral 
keys guide the amateur botanist to the name and description 
of the wayside flower. There is no reason why this codification 
and systematizing cannot take place, the structure of forms is 
really very simple, and the writer has no patience with the 
pernicious, though often unconscious, attempt to throw dust 
in the eyes of the layman and make technical problems appear 
.more difficult than they really are. 

He who has read with a broad comprehension the text to 
which this survey of statistical forms is the conclusion, will 


* A minor variation of both this and the next type of relative, is the chain-per- 
centage or link-relative, in which the base is not constant, but is always the preceding 
figure in the same series. It is a form of differential or successive differences. See 


rote in Chapter XXVI, page 307. 


THE STATISTICAL MATERIALS 683 


understand that graphic forms as well, are varieties and varia- 
tions of a common root illustration. He will know that this 
common root picture is the representation of a single number 
by astraight line. He will know that any collection of numbers, 
be it an abstract, geographical, or numerical distribution, can 
be presented graphically by a collection of straight lines, 
which, if joined end to end, form a 100% bar; but if placed 
side by side form a bar-chart. The development of the pie- 
chart from the 100% bar, he will understand as merely a sub- 
stitution of the circular for the straight line. The development 
of the curve he will understand as merely the connection and 
epitomizing of the bar-chart, suitable only for numerical series, 
whether frequency or historical. The varieties and modifica- 
tions of curves will have no mystery for him. The area and 
three-dimension charts will stand before him as amplifications 
and combinations of bar-charts and curve-charts, suitable for 
interlocking composite distributions. The map will be but a 
variant of the latter, in which two dimensions of the paper 
picture the independent variable in geographic distributions, 
by longitudes and latitudes. The 100% triangle, the nomo- 
graph, and the calculating charts will be but patterns in which 
the two dimensions of the paper are devoted to the laborious 
iilustration and proof of the very simplest propositions in 
geometry, This is really all there is to charts. 


Cuarter LV{il 


THE FUNCTION OF CHARTS 


A world turning to a saner and richer civilization will be a 
world turning to charts. From this conclusion, unwelcome as 
it may be, there is no escape. The case for the chart may even 
be sketched in a few schoolboy syllogisms, woven through the 
related ideas; civilization, clean-cut thinking; precision of 
thought, numerical statements; statistics, charts. With the 
last step in this chain, this book has attempted to deal. With 
a brief summary of statistical data the last chapter has provided 
us. There is no need to dwell upon the importance of precise, 
clear thinking, either in business or in economic studies. 
It remains to glance ahead a bit at the mechanics of the rela- 
tions which charts will assume with the civilized world at 
large, and to venture a few predictions as to the nature of 
these relations. 

And for this larger view it seems well to begin by amplifying 
our original definition. A chart is an image or graphic repre- 
sentation of abstract relations. Where these relations are not 
of a numerical nature, the chart is non-mathematical in 
character and is closely akin to the other graphic arts of a 
purely pictorial character; indeed its only distinction from 
paintings, photographs, and the like, appears to lie in the 
abstract nature of the ideas which it diagrammatically or 
schematically expresses. But where the relations are numerical 
and the subject of the chart is statistical, the chart is mathe- 
matical in character and forms a distinctly new branch of the 
graphic arts. While the artist will seek to present two groups 
of ten and twenty horses each by a picture of so many horses, 
placing his emphasis upon the realistic likeness of his drawing 
to horses, the chart-maker will seek to present the same 
objects by, let us say, two bars, which by their lengths express 
the numbers twenty and ten. His chart of horses will be 
exactly like his chart of two similar groups of ships, or his 

684 


a 


THE FUNCTION OF CHARTS 685 


chart of two very much larger groups of horses in which the 
group proportions are unchanged. He can, indeed, with equal 
facility make a chart for groups of two million and one million 
horses, a task which would be beyond the powers of the artist. 

In subject-matter, then, the chart is universal, and hence, 
too, in its potential appeal and usefulness. No one can think 
of two numbers and attempt to comprehend their significance 
without, at least unconsciously, visualizing them; the number 
which does not conjure up in our minds some picture of quan- 
tity remains meaningless to us. 

In this sense, therefore, everyone who deals with numbers 
is already a chart-maker and a chart-user. We have no choice 
between the use of charts and the use of statistics; we have only 
a choice between the use of written or physical charts and the 
use of imagined or ‘‘mind’s-eye” charts. Often, indeed, the 
latter are sufhcient, and many persons, it is true, still prefer 
under all circumstances to carry all the pictures of their num- 
erical data in their minds. But for the careful study of import- 
ant figures, or for the casual study of large bodies of important 
figures, this is obviously the less efficient method, and the 
physical record, the written or graphic chart, comes into 
service. It is more permanent, more convenient, and more 
accurate. 

The technique of the chart is also, in a sense, wider than 
that of the other graphic arts; indeed, it comprises something 
of the technique of all the arts. The reader of this book has 
seen that we have drawn statistics with pictures, and sculpted 
them as models and we have reproduced them by photo- 
graphy and by lantern slides and by printing. In this we 
have freely used design, relief and color. It may not be too 
much to add that some day we shall set charts to music, to 
enhance their graphic value, evolving a musical expression 
of statistics. This will seem less improbable when we consider 
its use in the accompaniment of moving pictures of charts. 

The animated chart, made possible by the motion-picture 
film, has long been a dream of the author. Its graphic value 
will be great in the presentation of fundamental economic facts 
to the general public, or of special statistics to special audiences. 
By its means the important chart can be presented in various 
stages of completion, and attention can be focused in turn 
upon each change, development, or addition to the picture. 
Thus in a bar-chart, the labels can appear first, then each bar, 


686 CHARTS AND GRAPHS 


with its data, can appear, one after the other, until the bar- 
chart is completed. Curves can be shown wiggling across 
co-ordinate rulings, with close-ups of each important added 
wiggle. Maps can appear first in outline and the shadings 
can appear and spread across the map by simple tricks of 
photography, and these shadings can be altered to show 
changing conditions for: successive points or periods of time. 
The “movie’’ of statistics is clearly coming, for schools and 
colleges, for the general public, for the scientific or academic 
meeting, and in business, for director’s meetings, for sales 
conventions, and for advertising purposes. 

Jn all chart-making, a distinction which will become in- 
creasingly recognized is the distinction between charts for 
popular consumption and charts for research purposes. This 
is no more than adapting the chart to the audience for which 
it is intended. And there can be as many different proper — 
charting ways as there are different degrees of familiarity with _ 
charts and ease in chart-reading. For extremely popular 
presentation, the pie-chart is always effective; bar-charts 
should be converted into series of circles and curves into 
vertical bars whenever possible. For more sophisticated 
readers the amount-of-change curve can be used; for the tech- 
nical and semi-technical, the simpler forms of the rate-of- 
change curve are permissible. The probabilities and other 
special projections will be really understood only by the 
experts; and are essentially charts for internal consumption 
in the research laboratory. 

Though everyone can be told how a bow is carried across a 
violin string, we do not expect all to play the violin well. 
And though the technique of chart-making can be simply 
explained, we cannot expect everyone to make good charts. 
The chief source of good charts will always be the statistical 
departments of large organizations. When the organization 
is an institution for the promotion of research in some special 
field, the statistical staff will of course be well manned. But 
the greatest strides, at least in chart- making, if not also in 
atictionl methods, will in the future be made in the statistical 
departments of large business organizations. 

In business the function of the statistician is two-fold, 
comprising on the one hand special research and investigations, 
and on the other hand, the co-ordination and intelligent report- 
ing of current business operations. In both of these, charts 


| 


THE RONCTION OF CHARTS 687 


are essential implements. In the research field the statistician 
has often a scouting function, his job being to look ahead and 
try to forecast the future development of the house and its 
markets. In the reporting field he assembles and interprets 
the operations of all the other departments; purchasing, pro- 
duction, shipment, warehousing, sales, and other collections; 
and of the business as a whole: inventories, costs, and profits. 
His position here is that of liason or intelligence officer between 
the responsible head of the business and his subordinates, and 
also between the responsible subordinates and their depart- 
ments. To get the fullest use of the expert intelligence in 
visualization and analysis, one of the vice-presidents may be 
himself a professional statistician and chart-maker of the 
highest specialized training, but in the past the average statis- 
tician has not often displayed a sufficiently practical view- 
point to justify this connection and the wealth of significance 
which lies in the records of the individual business house is 
untapped by those who must guide it. 

Comparable to the lawyer who brings to the guidance of 
business enterprise an intimate knowledge of legal technical- 
ities, is the business statistician who brings to it an intimate 


knowledge of statistical interpretation. In business houses 


where the operations and problems are of astandardized nature, 
his skill will not, except in very large concerns, be constantly 
needed and the statistician here becomes a consulting expert 
rather than a permanent officer. In such concerns the report- 
ing procedure can be quickly set in motion and standardized, 
so that it can be carried on thereafter by clerks. The Gantt 
progress-charts and a few of the simpler curves and maps are 
all that need be installed, after the proper system of records 
from the accounting and other departments have been estab- 
lished. In business concerns of more variety of operations, 
the trained statistician is necessarily more of a permanent 
member of the personnel and the work of forecasting is likely 
to be seriously entered into. Here the widest variety of charts 
come into use, for a nice understanding of their graphic value 
and true significance is available. Here it is often profitable 
to maintain a special statistical “laboratory” with complete 
facilities for statistical sources, compilation and analysis and 
for graphic records. 

The well-furnished statistical department should, of course, 
contain the mechanical calculators, the double-entry adding 


CHARTS AND GRAPHS 


Draftsman 


OT 


Typist 


= 
iJ 
Ho 
bes 
an 
= ( ’ 3 
2 
= ° 
> a 
2 CONFERENCE ROQM % 
te] os 
> 3 + 
2 
& 
FLOOR-PLAN 
for . 
a J 
Small 


Statistical Department 


Bereen over Chart Board 


KEY 
P © Projector 
& = Calculating machine 
A = Adding and listing machine Vice-President 
T = Typewriter, one long carriage 
one variable typ 
D = Desk 
L = Light-box 
F = File 
8 = Shelves 
M ® Map and tracings file 


Fig. 496. 


and listing machines, the multiplying and dividing machines, 
and perhaps the card-punching multiple-entry machines (the 
true posting machines). It should contain full drafting 
facilities and perhaps also blue-printing or photostating ma- 
chinery. These things obviously belong to the workroom, 
which should be apart from the office of the directing and 


THE FUNCTION OF CHARTS 689 


_ creative statistical officer. But the department is not complete 


without full accommodations for the successful study of its 


results. There should be a conference room, convenient to 


_ and properly fitted for the use which will be made of it by the 


directors or vice-presidents and responsible heads of the 
business. 


In the conference room the charts perform their chief 
function in business, as guides to the formation of policies. 
The room should be equipped with a light-box for the com- 
parison of curves and with a screen and projector for charts 
which it is desirable to exhibit to several persons at once. 
Important data can be permanently posted on large wall- 
boards and these wall-boards, by the use of sliding panels, 
can hold large bead-maps as well. All important data should 
be on record in chart form, either in looseleaf binders or vertical 
files. Needless to say, the room should always be locked up 
when not in use and the keys to it should be in the hands of 
but two or three persons. It should be the repository for all 
information about the concern which is of value in the forma- 
tion of policies, this information being in chart form because 
of the ease with which it can then be consulted. 


Of a much more general nature are charts for popular con- 
sumption. ‘These are appearing with increased frequency in 
newspapers, general magazines, and technical publications. 
The day will come when no statistical compilation will be 
regarded as complete until it is illustrated with charts which 
present its major significance. The greatest development of 
charts, here, however, will take place in the advertising 
columns, and in general for propaganda work. For the proper 
chart is an excellent weapon against the inertia, indifference, 
and often hostile attitude of the average reader. It is not 
merely the best kind of eye-catcher for calling attention to 
numerical data, it is also the most convincing proof of that 
data. The most casual reader stops a moment before any 
diagrammatic puzzle to examine it. If he finds incidentally 
that he immediately understands it, he is perhaps at once 
pleased with it and is sure at least to carry away with him a 
memory of the message it conveyed. ‘That such charts should 
be of the simplest, goes without saying; and here too, expert 
skill in chart-making is desirable. For the right chart 1s 
strong in inverse ratio to the technical ability of the reader, 


690 CHARTS AND GRAPHS 


> 


and the less effective would be text or tables, the more powerful 
grows the right chart. 

In all fields, scientific, academic, and commercial, the chart 
is a medium of expression too forceful to be overlooked, too 
valuable to be neglected. Its future growth will assuredly be 
rapid and perhaps in many ways even startling. In this book | 
we have set forth many ways for the presentation of statistics 
and statistical relations. The category is, however, by no 
means complete. It cannot be complete, for the charts are 
still in the making, and the methodology of graphic illustration 
is in no sense that of a perfected art. There is room for much 
improvement in existing chart-forms as well as in the develop- 
ment of altogether novel forms. New ideas will come out of 
the research laboratories, new methods, new forms, new 
charts. The distinction between graphs for popular publica- 
tion to the general public and graphs for internal consumption 
in the statistical workshop, will become more marked; and as 
public knowledge increases, charts will pass out of the work- 
shop into the magazine and book page, no less through adver- 
tising than through text columns. 

We are finding a new language, the grammar of which 1s 
not yet completed, nor the dictionary written. It is well that 
this is so, for codification and systemization easily bring 
stagnation; and volumes such as the present, in which the 
existing material is set in order, must not be allowed to stifle 
new growth. The reader is urged not to permit the rules laid 
down in this book to restrict his efforts, but rather to allow the 
principles set forth tu stimulate his imagination and enterprise. 
The pictorial display of mathematical and numerical state- 
ments 1s an illustrative art, with the high object of facilitating 
human understanding and vision, an end the achievement of 
which justifies all means, be they orthodox and accepted or 
novel and previously untried. 


FINIS 


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APPENDIX A 
IMPLEMENTS FOR MAKING CHARTS 


The equipment which should be available in a chart- 


_making or statistical office depends, of course, a great deal 


upon the types and forms of charts which will be developed 
and the nature of the data which will be handled statistically. 
Charts can be made at home or in the very smallest office, 
with nothing more than a draftsman’s ruling pen, some India 
ink, and good paper. As the chart-making work grows, more 
drawing pens will be added in order that different colored inks 
can be ruled in without delays, and in order that several oper- 
ators can be drawing at the same time. A good drawing board 
is necessary and perhaps two drawing boards are best, one 
about 24 by 30 inches for large charts, and the other about 
18 by 24 inches for smaller charts. Together with the drawing 
board, there should be a T-square and small triangles or tri- 
squares. Occasional need arises for one or two patterns of 
French curves. A good drawing set includes a compass or 
dividers for the drawing or circles of circular outlines. Several 
dotting machines for the drawing of dotted, broken, and dotted 
dash lines are on the market, and when they can be success- 
fully used, save considerable time, but they are not always 
successfully used. A section-liner is almost essential for much 
cross-hatching work. A protractor is necessary where circles 
will be divided into proportional parts and angles must be 
used. For bar-charts, a double ruling pen, or railroad pen, is 
a great convenience, adding to the appearance of the chart 
by making more absolute the uniformity of the bar widths, 
and greatly decreasing the amount of time required for the 
making of the bar-chart. (This pen rules in two parallel lines 
simultaneously.) Bar-charts can also be made on large scales 
with adhesive tape or passe-partout; and several mechanical 
bar-charts are now on the market in which cloth tape is un- 
wound from invisible or hidden spools and drawn out to the 
691 


692 ' CHARTS AND GRAPHS 


required length of the bar-chart. These last are useful where 
the length of the bar-chart must be frequently changed and 
brought up to date (as in a sales-manager’s office where the 
bars represent the weekly averages or cumulatives of the work 
of the individual salesmen). In map work, a curved ruling pen 
is sometimes an advantage for the drawing of rounded curves; 
better still, is the Payzant lettering pen. A planimeter is 


ee | 3 3 ¢ Ss 
Permission of Keuffel & Esser, N. Y. 


Fig. 497. The Payzant Lettering Pen. 


Patented. 


No. 000 


often useful for checking up total areas on the map. A panto- . 
graph is a device by which outlines can be copied on larger or 
smaller scales, and is sometimes useful in map work; very 
cheap pantographs can be obtained, which will ordinarily be 
satisfactory. 

A straight-edge is useful for the cutting of paper, and the 
best knives available are the ordinary one-sided razor blades, 
as the paper-cutting knives require frequent sharpening. The 
straight-edge is necessary because from time to time in the 
cutting and trimming of paper the edge itself will be cut into 
by the knife and if the T-square has been used the T-square 
will then be ruined. Ordinary camel’s-hair brushes are often 
used for applying water-color or ink to maps; but the best 
way of coloring maps, as has been previously described, is 
by the use of wax crayons, the cheaper and waxier the better, 
the wax being afterward removed by a sharp knife-edge or 
razor blade, leaving the desired color tint. For the quick 
filling in of bars and solid areas, the lettering pen is desirable. 
It can be obtained in many sizes, but the results are not 
entirely even and smooth unless the operator is skilled. A 
special pen (the Payzant) has been put out for fine lettering, 


| a 


i 


IMPLEMENTS FOR MAKING CHARTS 693 


which has a round nub with a well similar to that in a drawing- 
pen, holding considerable ink. This pen is also available in 
several sizes. 

Typewriters should be used as far as possible in the letter- 
ing of charts, as well.as in the entry of data upon the charts. 
The process is much more rapid than hand-lettering, and the 
results are, for the average-sized chart, usually better. There 
are two sizes of standard typewriting type, “pica” (10 char- 
acters horizontally to the inch) and “elite” (12 characters 
horizontally to the inch). Unless large type is especially 
desired to facilitate small reproductions, the elite is better, 
for it enters all figures in less space. Any special type-faces, 
such as Gothic or Italic, may be had, but the usual type-face 
is Roman. The figures come in two styles in all machines, 
“book-keeping” and “regular.” Book-keeping type has 
slightly greater visibility but gives uneven lines, as the 
numerals have swinging tails. The regular numerals are 
usually more satisfactory. 

The typewriter carriage to use for chart-making should 
accommodate I1l-inch paper (and larger, if charts are being 
made on sheets larger than 84x11 inches). One standard 
machine (the Royal) will take 11-inch paper on its regular 
carriage, all other makes require long carriages. The type- 
writer to use for chart-making should also so hold the paper 
as to print down to the very bottom edge of the paper without 
shifting. There is but one standard machine (the Royal) 
which will do this. Besides the standard machines, the 
Hammond has good features for chart-making in that it will 
print any style of type at a moment’s notice, 9 lines to the 
inch instead of 6, and will space the characters properly (and 
if desired, as close together horizontally as 18 characters to 
the inch); but this machine is not so easily handled by most 
operators as the standard machines, because it has a three- 
shift key-board, and, while it will accommodate any size of 
paper, the paper is likely to shift slightly in it and cannot be 
used down to the bottom edge. 

For the statistical work, special computing machinery 1s 
desirable, both for its speed and for its accuracy, being for these 
reasons, where the amount of statistical work to be done is 
considerable, a great economy. Computing machinery is in 
general of two types, the adding machine and the calculating 
or multiplying machine. The adding machine can be obtained 


694 CHARTS AND GRAPHS 


in both printing (or “listing’”) and non-printing types. Ob- 
viously, the printing machine is far more desirable because the 
operation can be checked back by an examination of the printed 
page or record. A special type of adding machine, called the 
duplex model (Burroughs), is desirable for work in which 
totals of parts will be required. These part totals are known 
as transfer-totals, the machine having two faces, the one of 
which clears without removing the record from the other face, 
so that several adding operations can be conducted at the same 
time. The duplex machines are particularly useful in the com- 
piling of statistics by States, the part totals being taken off 
for State groups, and afford a great saving in the detection of 
errors when the data is checked over. Still a third type of 
adding machine is the mechanical tabulator (such as the 
Hollerith), which works with punched cards in which the 
amounts to be entered with full descriptive detail, are repre- 
sented by holes in a card, and these holes operate the machine, 
just as do the holes in the records of the player-piano and 
similar devices. This is the true posting machine—it sorts, 
posts, and adds, with a typewritten record if desired, auto- 
matically. 

The calculating or multiplying machines so far manufactured 
are only of the non-print type and leavenowrittenrecord. This, 
of course makes errors more difficult to detect. There are two 
general types of machines. ‘The first (such as the Burroughs or 
Comptometer) automatically adds as fast asthe operator punches 
the keyboard, and calls for specially trained operators who can 
punch several keys at once and continue punching the same 
keys the proper number of times to effect the multiplication 
(shifting the position of their fingers on the keyboard for 


each new digit in the multiplier): The second type of machine © 


performs the same operation automatically from a single 
punching of the keyboard. The operator may be required to 
turn a handle the proper number of times to effect a multipli- 
cation, and to shift the recording dials one space for each digit, 
but the work is very quick and absolutely accurate. In the 
latest model German machines, electrically driven, the work 
is entirely automatic after the setting of the keyboard. Adding 
and calculating machines are economical of time and expense 
if electrically driven. 

All of these calculating machines are virtually multiple- 
adding machines, for effective quick addition. They are 


| 


IMPLEMENTS FOR MAKING CHARTS 695 


accurate to the last figure recorded. For many statistical 
purposes, however, such as the figuring of percentages, accur- 
acy is not necessary beyond the third or fourth figure. For 
such work, a slide-rule is sufficient and, though harder on the 
eyes, is much more portable and is indispensable in the statis- 
tical office. The accuracy of reading increases with the length 
of the rule; and slide-rules are made five, ten, and twenty 
inches long. Further accuracy can be gained by the use of 
magnifying glasses fitted on the runners of the rule. Needless 
to say, slide-rules will perform many operations outside of the 
powers of the calculating machines. 


APPENDIX B 


STEPS IN MAKING CHARTS 


In statistical offices, both large and small, it is desirable 
to have as near as possible an approach to what may be called 
“straight-line methods” of chart production. Only by the 
institution of such methods can the routine work of a large 
number of charts be satisfactorily and economically accom- 
plished. For this purpose, it is desirable to break up the work 
of chart-making into various steps and stages, and to have the 
individual charts, as they pass through these stages, pass from 
the desk of one operator to another in a regular series and 
direction. The various steps outlined below will be found to 
be a fairly complete list of the stages through which different 
kinds of work will pass. Very often, however, some one or 
more of these stages will be omitted for particular charts and 
for particular data. 

I. The first step in chart-making is, of course, the collec- 
tion of data or statistics, namely, the information to be shown 
upon the chart. This information can be gathered in two dif- 
ferent ways. The first way ought, wherever possible, to be 
followed out, whether the second is used or not. 

The first way to gather data is to consult and collect all 
information previously compiled by other investigators on the 
particular subject. ‘This is a class of research work ordinarily 
involving the consultation of the various books, pamphlets, 
and other authorities in the public libraries, and calling for 
the services of fairly skilled library workers. A reasonably 
complete knowledge of the various sources and authorities in 
which the particular information sought is likely to be found 
in its most useful and complete form, is of course desirable, 
in order that the search will not take too much time. When 
the information has been found, it can be carefully copied 
upon specially prepared work-sheets or data sheets by the 
investigator, or, if the data is in compact form, it cah 

606 


STEPS IN MAKING CHARTS 697 


be photographed or photostated and the photostatic copies 
used in the office. The latter method, though apparently 
more expensive, is far more accurate and reliable and more 
economical of time, so that, in the end, it is generally the 
cheaper process. 

The second method of gathering the information is by the 
use of field investigations. It is an independent and original 
study for the purpose of securing primary information rather 
than secondary information (1.e., information taken second- 
hand from other investigators). The field investigation may 
be carried out by a skilled investigator, if it is not too exten- 
sive and if the resources are available for sending a thoroughly 
skilled investigator out. When this is not possible, or when 
the extent of the investigation is very great, the method of 
questionnaires can be used. In this case, the art of question- 
naire-making comes into play, for the drawing up of a question- 
naire is by no means as simple as it might appear. 

A good questionnaire is one in which’no question can be 
misunderstood; one in which each question is capable of only 
one meaning and of a precise answer of one type only. More- 
over, a person who has drawn up a questionnaire must be 
able, beforehand, to envisage his entire problem, foreseeing 
all moot points and issues which will arise in the course of the 
investigation, and to which answers will be desired. Lastly, 
the questions must be so framed as to avoid, as far as possible, 
any psychological reactions either upon the part of the investi- 
gator or of those whom he questions and consults for his in- 
formation. In fact, the psychological difficulties about many 
questionnaire problems are the principal obstacles, and the 
method of questionnaires is fast losing ground for precise 
statistical compilation, because of the careful analysis and 
psychological interpretation, translation, and correction to 
which the answers must be subjected before they can be satis- 
factorily compiled. And great as is the task of preparing and 
conducting a satisfactory questionnaire, the problem of 
compiling its answers is sometimes even greater, and requires 
a staff of more than ordinary intelligence. 

And in both research and field investigation work, wherever 
clerical tasks have been performed, it is, of course, necessary 
that careful checking be done on all such clerical work in order 
to catch and correct errors which are humanly inevitable. A 
definite place in the schedule of preparing charts must be 


698 CHARTS AND GRAPHS 


given to this work of checking back for accuracy on all the 
clerical work performed. 

Il. The second step in the routine of chart-making is 
ordinarily called the computing. This often requires that the 
data be first copied upon specially prepared forms or work- 
sheets, so that it may be subjected to the proper processes. 
The computing should, as far as possible, be planned con- 
siderably in advance in order that work may progress evenly 
and smoothly. The work is of a clerical or statistical nature 
and calls for the services of statistical or computing clerks, and 
frequently also for a battery of adding and calculating 
machines. The work-sheets which are to be used for the job 
should be carefully designed with an eye to the machinery by 
which the processes of calculation can be most easily performed. 
Thus, in cases where totals and subtotals are desired, the work- 
sheets should be designed so as to fit into the adding-and- 
listing machines in order that the machine may operate 
directly on the work-sheet and not upon the usual tape. If 
the work is first done on the tape, it will have to be copied on 
the work-sheet, with the unavoidable percentage of error and 
the great additional time and labor involved. The only alter- 
native is to paste the tape on the work-sheet, and this will 
not be possible if the work-sheet is not large enough. Needless 
to say, for all computing steps the most rigorous checking for 
errors is necessary. If possible, the computing should be so 
done that it is self-checking, or easily checked for accuracy by 
a single checking operation performed upon the totals for the 
data. It is best, therefore, when the adding machines are to 
be used, to have work sheets in which the lines coincide with 
the adding machine lines, so that the sheet can be run through 
the machine instead of tape. Moreover, if possible, the 
items to be added should be entered in a column, with blank 
lines between them, so that the machine entries may appear 
immediately below the hand-written entries, this reduces error 
and facilitates checking. 

Ill. The next step is the beginning of the chart-making. 
The general character of the charts to be used should have 
already been determined. Suitable chart-forms should have 
been obtained from some publisher, or made to order by one’s 
own printer. These forms should accommodate the data in 
typewriting. And this step, therefore, may be called “entering 
up the chart.” It calls for the services of an intelligent and 


~~e 


STEPS IN MAKING CHARTS 699 


capable “tabulating typist,” for the work must be both accu- 
rately and neatly done. If, in the case, for example, of curve- 
charts, the ordinates at which the data must be entered have 
been placed at uniform typewriter intervals of one-third of 
one inch, the typist will be able to work rapidly and smoothly, 
and work will cross this desk promptly. One good typist, 
under such circumstances, can keep half a dozen compiling and 
drafting clerks busy, and will generally find time to do comput- 
ing work as well. Again, checking for errors is necessary. 

IV. When the final data has been entered upon the chart, 
the next step is one of plotting this data in chart form. The 
immediate proximity of the data to be charted upon the chart 
itself, as placed there by the last operation, makes this drafting 
or plotting process extremely easy, and where the chart-fields 
have been already printed or drawn upon the paper, the 
drafting requires little more than the proper selection of 
plotting points and the careful ruling in of curved lines or bars. 
The work calls for the services of an ordinarily intelligent 
clerk and only in the case of very complicated charts, or in 
cases where extra lettering and entering of data will be done 
by hand, is it necessary that skilled draftsmen be employed. 
The draftsman should work under the best available light, as 
the eye-strain of careful plotting or ruling is severe. Art 
school students and engineering school students are generally 
qualified to perform this work capably. Again, the process of 
checking must be carefully done to detect error. 

V. The final step on any chart'is the labelling and the 
finishing up of the various details left unfinished in the type- 
writing and plotting stage. Unless the form of title for the 
chart has been standardized, the problem of a correct, com- 
plete, and easily understood title for a chart is sometimes very 
dificult. And the title should, in general, be made by the de- 
partment head or statistician who is responsible for the work. 
There should always be a portion of the chart sheet in which 
the title can be conveniently located where it will be at once 
apparent to the reader. In the case of historical curve-charts, 
where the field is low on the page, the title naturally belongs at 
the top. In other cases where the chart 1s higher up on the 


- page, the title can be placed below the chart. 


A good title should not merely give the nature of the 
phenomena shown by the chart together with the general kind 
of analysis followed by the chart, but it should also give such 


700 CHARTS AND GRAPHS 


distinctive details as will separate the chart from all others in 
the series. Where the series of charts is clearly connected, 
as in historical curves, the date or year of the individual chart 
can be placed in the corner of the paper, the chart title being 
reproduced alike on all charts in the series. In addition to 
the title of the chart, the chart should also tell its source, that 
is, the authority for the information it presents. And, in 
addition to these two items which must be typed upon the page, 
there is generally considerable labelling of data columns. 
Sometimes there is labelling of individual curves upon the 
chart-field itself. 

When all this has been done and the chart has been dressed 
up in its final form, it should be carefully inspected by some 
one competent, as far as possible, to detect errors which appear 
in the chart; and the chart should have in one corner a place 
for the “O. K.” signature of the person in authority who has 
finally approved of the chart. In addition to this approval 
signature, the data of approval should be shown so that charts 
of different date of manufacture can be easily seen and the 
latest and most reliable chart distinguished. 

Personnel.—In short, it will be seen that the statistical 
office has, in the main, five major processes, namely: research 
work, computing, typing and printing, drafting, and inspec- 
tion (including titling), and, in general, it may be said that 
these processes call for different types of workers, namely: 
‘research workers or librarian, statistical clerks, typists or 
letterers, draftsmen and artists, and statisticians. 

Notes.—In addition to the finished chart itself, it is some- 
times desirable to have appended notes or explanatory com- 
ments which will serve to interpret the significance of the 
chart to the reader or executive who will consult it, and which 
will point out to him the important facts displayed by the 
chart. The writing and composing of these explanatory notes 
should be composed by the statistician in charge. These 
explanatory notes should be in the most easily comprehensible 
form. ‘Their composition calls for an extremely practical 
understanding of the point of view of those who will read and 
consult the chart, and requires a return to the language and 
to the non-technical line of thought which will be pursued by 
the layman. 

Reports.—Nothing has been said here about the mobilizing 
and assembling of a large number of charts upon one subject 


om re a: 


STEPS IN MAKING CHARTS 701 


into the form of a single coherent report. This is usually con- 
sidered a matter for the skill and judgment of the statistician 
himself. According, however, to so excellent an authority as 
Mr. Charles P. Steinmetz,! there are three kinds of reports, 
and the most complete report generally contains these three 
types within itself. The first is the general report in which 
the final conclusions and significances are summarized briefly, 
the report perhaps taking up about 10 per cent of the entire 
report. It is this part of the report, and often this part only, 
which will be read by many executives, or the average reader. 
In the second part of the report, these conclusions which ap- 
peared in the summary or general report are expanded in 
greater detail to show their bases or foundations and to enable 
the careful reader to delve deeper into individual phases and 
aspects. This second part may contain about 30 per cent of 
the total number of pages in the report. The third part of 
the report is the technical authority and technical detail 
which will be read only by those who are extremely anxious 
to check up upon the work of the compiler, either for the sake 
of repeating or elaborating the investigation, or for the sake 
of detecting errors or confirming the accuracy of the informa- 
tion given. This part may often take up the greater portion 
of the report. 


1 Steinmetz, Charles P., Engineering Mathematics, McGraw-Hill Book Co, 1917, 
p. 290-293. 


APPENDIX C 


METHODS OF PRESENTING CHARTS 


In the main, the charts described in this book have been 
discussed upon the basis of presentation upon ordinary size 
letter paper, that is, paper measuring 83x11 inches. Occa- 
sionally, larger sheets of double this size have been mentioned, 
and in the section on models the need for mounts and con- 
tainers of uniform size has been discussed. The 83x11-inch 
paper is perhaps the most generally convenient because of its 
conforming to standard sizes of office paper and vertical and 
other filing methods. In an office where legal size paper is 
used, it would obviously be better to adopt sheets 84x13 
inches as the standard chart size, and on occasion, to use 
sheets of double this dimension. 

In the section on curves, which form the great majority 
of charts, the desirability of positioning the curve in one 
corner of the paper for ready comparison and of leaving large 
margins above and to the left of the chart-fields has been 
explaified. The chart-form itself should be carefully designed 
as the one most suitable to the type of chart-work which will 
be done. 

It is of the greatest importance that the ‘‘field,” or rulings, 
of the chart-form be in a faint ink, preferably green or gray. 
(The orange and reds are hard on the eyes; the blues will not 
photograph.) Whenthe charts are to be reproduced on a much 
smaller scale, it is well to make all but the more important 
' co-ordinates in blue, so that they will not be reproduced. For 
this purpose, blue co-ordinate paper can be wsed, the co- 
ordinates which should be reproduced being ruled in by hand 
in black ‘ink. 

The usual types of maps are not ordinarily printed upon 
standard sizes of paper, and if a great deal of map-work is to 
be done, it is well to have special maps printed upon regular 
sizes of paper to conform to the rest of the charts in use. Maps 

702 


METHODS OF PRESENTING CHARTS 703 


on the 83xll-inch paper, often of inferior quality, can be 
obtained from a few manufacturers of charting materials. 
These are sometimes better than more elaborate maps, as 
they are generally only outline maps showing State or county 
boundaries. 

Attempts have been made to establish standard charting 
forms upon cards for card-catalogue filing, small curve- 
charting fields being printed upon 4x6-inch cards in one corner 
of the card. The fields are ruled aff for amount-of-change 
curves only, with ordinates for 52 weeks, 12 months, and 31 
days, in the usual way for historical curves. The amount of 
data which can be entered upon these card charts is, of course, 
limited and the detail in which the curves are shown is not 
great. The thickness of the card prevents, to a certain extent, 
the facility of “light analysis.” Some publishers present 
these small charts on thin paper suitable for tracing or “light 
analysis,” as part of a loose-leaf note-book system to be 
carried about in one’s pocket. 

A major problem in graphic presentation arises when the 
charts are to be shown to large audiences. The practice is 
often followed of making the drawings extremely large, say 
three by four feet in size, on heavy paper which can be unrolled 
and pinned against a bulletin board for display. These rolls 
of paper, however, are difficult to carry about and are easily 
damaged. This is even more true when heavy card-board is 
used, which can not be rolled but must be carried about flat. 
The best advice appears to be to present the chart upon 
tracing cloth, which can be fastened at one end upon a large 
stick of wood and easily rolled or unrolled. Where expense is 
no consideration, the window-shade rollers on which school 
maps are mounted can be used. These are contained in long 
narrow boxes to keep the chart dust-proof, the chart being 
unrolled by pulling the lower end out of the box in the same 
way that a window shade is drawn down. 

Much the best method for the display of charts of large 
size to an audience is by the use of lantern-slides and a lantern 
slide projecting machine. These machines can be purchased 
for small sums in a very handy shape, folding up like valises 
and easily portable. The lantern slide can be made by any 
photographer at small cost, from the orginal chart used in 
the office made in the usual way upon ordinary size paper. 
Where colored areas are shown upon the chart, the same colors 


704, CHARTS AND GRAPHS | 


may be shown on the lantern slide, by coloring the slide with 
Japanese transparent photographic colors. This coloring work 

will be done by the photographer according to instruction, or 

according to the original chart which has been photographed. 

Lantern slides are smaller than post-cards and easily carried 

about. Except through breakage, they are not easily 

damaged. 

The lantern slide method has more to recommend it in the 
fact that a drawing which will be seen upon the lantern slide 
will easily be visible to the entire audience. Where the very 
large original drawings are used instead of lantern slides, the 
lines in the drawings have to be made very much thicker to 
make them visible, but with lantern slides, the faintest lines, 
if distinctly visible upon the plate, will be clearly projected to 
the entire audience. A safe rule is that whenever the original 
office copy of the chart, from which the lantern slide 1s made, 
is larger than 84x11 inches, the lines upon it wili not be clear 
and definite upon the lantern slide (because of the reduction 
in size) unless the lines are made heavier; but in the case of 
originals up to 84x11 inches, not only will the ordinary 
markings be clear and distinct but the ordinary typewritten 
labels and data will also be visible. 

A very expensive machine has been devised, which is 
useful in large offices for the display of many charts to board 
meetings and other small audiences. It is a projecting ma- 
chine which does not require lantern slides, but which will 
reflect the image of the original chart upon the screen. The 
convenience of this type of machine is, of course, very great, 
as it eliminates the delays and expense of lantern slides, and 
permits the exhibition of any material at a moment’s notice, 
even though the need for such an exhibit had not been fore- 
seen. The machine is, therefore, valuable where a large num- 
ber of charts may be shown and it is not certain beforehand 
which ones will be desired. 

The method par excellence for popular audiences is the 
moving picture film and machine, with the charts shown as 
actually developing and building up. The manufacture of such 
films is not easy and is very expensive, but the results fully 
justify it. Thus, a bar-chart shown in this way would first 
appear merely as a list of items, and then, one by one, the bars 
would appear upon the screen, until the entire chart was as- 
sembled. Such a chart receives careful study and all its parts 


are understood, and their significance is grasped by the audience. 

‘The reproduction of charts in large numbers involves special 
problems. Where only a few copies are desired, photostating 
is the best method. Blue-printing is a more economical 
method but its results are neither so clear nor so attractive. 
By the use of blue-printing, only negatives can be obtained, 
in which black areas appear as white and white areas appear 
as blue. A somewhat similar method is known as the Van 
Dyke process, or black-line and brown-line process. These 
are obtained partly by offset and partly by photographic 
methods, and positives as well as negatives can be secured, 
but the original chart should be upon very translucent, un- 
water-marked paper (best of all, upon tracing cloth) and very 
distinctly and clearly drawn. In these two methods, the 
process 1s a photographic “printing-through”’ one, the light 
passing through the chart to a sensitized surface. It is 
necessary, therefore, that no extraneous matter be upon the 
reverse side of the chart and that no corrections be made upon 
the chart by overlaying fresh paper or Chinese white. It is 
also desirable that the chart paper be clear, un-water-marked, 
translucent. In these respects, the blue or sepia print, and 
the black-line or brown-line print are unique. As they are 
“printing-through processes,”’ it is always advantageous, when 
there is typewriting upon the original drawing, to back up the 
sheet with reversed carbon paper, so as to get an additional and 
coinciding imprint of the typewriting upon the reverse side of 
the original. 

The above limitations and precautions do not apply to 
the truly photographic processes, that is, the photostat, the 
photograph, and the photo-engraving. In these, the light is 
reflected back from the surface of the chart to a sensitized 
surface, without passing through the chart. Thick, opaque 
paper can be used, with corrections in Chinese white or on 
special slips of paper pasted over the incorrect parts; also, 
the condition of the back of the chart does not matter. When 
only a few copies are needed, photostating is the best process, 
far more convenient and only slightly more expensive than 
blue-printing. The least expensive course is to get a photo- 
static negative and as many blue-print positives as desired. 
When sufficient copies are desired to make printing advisable 
(that is, printing from metal plate) it is necessary to use either 
the “line-cut” or the “half-tone.”’ 


METHODS OF PRESENTING CHARTS 705 


706 CHARTS AND GRAPHS 


The line-cut is the more economical but shows only full 
black and white markings. The half-tone (in which the 
object is photographed through a screen) 1 is a more expensive 
process, giving results with all variations and tints of grey as 
well as almost full white and full black. Half-tones are much 
more expensive than line-cuts and require more time in their 
manufacture, but the results, of course, are better when areas 
are shaded with different tints and colors in the original chart. 
The line-cut is sufficient for the ordinary bar-chart or curve- 
chart and can often be given elaborate shadings and cross- 
hatchings by the use of the Ben Day process. 

For all photographic methods, including printing, care 
must be taken in the choice of colors, when colors are used 
upon the original chart, for as has been previously pointed 
out the various colors reproduce differently by photography 
than would be expected from their appearance to the eye, 
and certain shades which appear decidedly different to the 
eye, may be exactly similar to the camera and reproduce alike. 
Red, of course, photographs as black (that is, appears as black 
upon the photographic print), while blue does not photograph 
at all but appears as white upon the photographic print, and 
yellow appears almost black. Thus it will be seen that a scale 
from red to blue arranged in chromatic sequence of colors will, 
to acertain extent, photograph as a natural sequence of grays 
ranging from solid black to white in the photographic print. 

Three other methods of reproduction are in ordinary com- 
mercial usage, the hectograph, the mimeograph, and the 
multigraph. ‘The first two of these can be used satisfactorily 
for the reproduction of charts. In the hectograph, the old- 
fashioned jelly-offset process, special hectograph ink can be 
used in several colors which will simultaneously reproduce, 
the copies appearing somewhat paler than the original but 
reproducing color for color at a single offset. The number of 
copies that can be secured from hectographic offset is, however, 
limited. The best offset processes claim to afford as many as 
50 to 70 copies from a single original, but as a rule 25 to 30 
are all that will be satisfactory. When not more than two or 
three dozen copies are desired, the hectographic process is 
extremely simple and satisfactory in the average office, its 
only requirement being that the special hectographic or copy- 
ing ink or pencil, or typewriter ribbon be used in the making 
of the original chart which is to be copied. 


_ METHODS OF PRESENTING CHARTS 707. 


The mimeographic process is essentially a stencilling one, 
the chart being first drawn upon a fine wax or fibre stencil 
and then laid over the drum of an inking and printing machine, 
the ink passing through the cuts in the stencil and printing 
. upon paper. The mimeographic process will produce as many 
copies as desired up to several hundred, but is limited, like 
ordinary printing, to one color only, for the color is deter- 
mined by the ink which has been previously placed upon the 
drum of the printing machine. The mimeographic process can 
be conveniently used where mimeographic reproductions are 
already being made of typewritten copy. Its limitations are 
that it shows only one color, and that it is impossible to repro- 
duce solid black areas on the mimeograph. Shading must be 
done by cross-hatching, and extensive cross-hatching is, apt 
seriously to damage the mimeograph stencil. Moreover, the 
mimeographic process may prove a dirty one, even for those 
who are experienced with it and certainly for the beginner. 
And it is extremely difficult to adjust the stencils upon the 
printing drum so accurately and so evenly that the reproduc- 
tion will be precise; as a rule, a slight curvature or wrinkling 
of the stencil is observable; this, of course, makes for irregular, 
curved, or broken lines on the chart when straight lines are 
desired. 

There is a special instrument known as the mimeoscope 
which is well adapted to the drawing of charts upon mimeo- 
graph stencils. It is also useful in the office as a “light-box”’ 
for general light analysis. The machine consists of a strong 
electric light under a ground glass (ground to diffuse the light). 
By the use of this machine, charts can be easily traced on 
mimeograph stencils, using the various kinds of mimeograph 
styles provided for the purpose. The mimeograph method is 
adapted to the reproduction of simple charts in one color only, 
without solid areas, and not requiring precise reproduction, 
when the number of copies desired is between 50 and 500; 
for a smaller number of copies, the hectograph processes are 
satisfactory, and for a larger number of copies the printing 
processes are more economical. 

In a few cases where charts are largely prepared upon the 
typewriter, and not more than two, three, or four copies are 
desired, the carbon copy method of reproduction can be used. 
Carbon paper can be obtained in red as well as black, and some- 
times in other colors, and the first two or three carbon copies 


708 CHARTS AND GRAPHS 


are likely to be very good when the paper on which the charts 
are made is not too thick or soft. Sometimes the labelling and 
typing for a number of copies can be done at one operation on 
the typewriter with carbon paper, or in two operations when 
different colors of carbon paper must be used (first printing 
up all of one color and then substituting the other color of 
carbon paper and printing up that). Drafting upon the charts 
must be done on the various copies independently with 
drawing pens in the usual way, making each of the copies 
original so far as the drafting is concerned unless, as in the 
case of bar-charts, the chart can be made with the type- 
writer. Bound carbon sheets (in binders) help to prevent 
shifting of the copies. 

Whenever typewriting is done for charts which are to be 
blue-printed, or photographically copied by direct printing (in 
which case, the light will have to pass through the entire 
chart-paper), it 1s best to insert a piece of carbon paper behind 
the original chart and print a reverse copy upon the back of 
the paper to add intensity to the typewriting on the chart. 
Only fully inked typewriter ribbon should be used for charts 
which are to be photographically copied, as it is desirable that 
the printing should be as intense as possible for the best 
photographic results. 


eo 


Aprenpix D 


COLORS IN CHARTS 


A great deal of controversy takes place over the use of 
colors in charts. In a previous appendix we have discussed 
those limitations to the use of colors which arise from the 
needs of specified reproduction processes. In this appendix 
will be considered the use of colors in cases where no mechani- 
cal limitations are imposed and colors can be judged entirely 
upon their own merits. 

The argument against colors takes the line that the average 
business man is not accustomed to them, and will consider 
more carefully a product in the familiar black and white. 
The colors in this view tend to make the chart “pretty”? and 
prettiness is rightly to be avoided. Other things being equal, 
artistic effects are always desirable, but never to the point 
where they attract attention and distract the reader’s mind 
from the message of the chart. 

But even in this view, not all lines need be equally strong 
and it is freely conceded that the co-ordinates of the field of 
a chart should be as light as possible, and, even better than 
in black, ruled or printed in gray ink. For the field is only the 
background, and the lighter shade throws the curve or plotting 
more distinctly into the foreground. And the best practice 
has already gone further and given to the field of a chart 
invariably the color of green. The green should be medium 
light with somewhat more yellow than blue, so that it will 
photograph as well as gray. The advantage of the green 1s 
that no confusion with black plotting or lettering is possible. 
The rule may be extended to make it universal, even the 
maps to be used in statistical reports being printed in green 
outlines. 

Red is a color to which the accountant is accustomed to 
give a negative significance, and may well be used for con- 
trast with black in curves and map-shadings, wherever it 


709 


ye CHARTS AND GRAPHS 


applies to opposite data. Thus on a map the red should be 
used for unfavorable conditions, and in curves for costs or 
expenses, and the like. It is one of the great advantages of 
the red that the corresponding data (and scales, if any) can 
be typewritten in the same color without difficulty when the 
typewriter is equipped with a bi-color ribbon. Brown, green, 
and blue typewriter ribbons can also be secured, but require 
special shifting of ribbons in the machine. Red and black 
are the main colors used in the charting office. 

Blue is taboo in chart-work and should rarely be used. 
Its least fault is that it is hard on the eyes when used steadily 
in the place of black. Its great fault is that it does not photo- 
graph, and will therefore disappear if the chart is photostated 
or blue-printed or photographed in other ways. In maps it 
may be used for shading to indicate favorable conditions, 
with the understanding that when photographed, these areas 
will be white; in curve work it may be used for lines, or rulings 
which are intended to disappear when photographed, either 
for secrecy or to eliminate details useful only in plotting and 
undesirable in reduced copies. When blue is used to disappear 
either in ink or typewriting, care should be taken to see that 
it contains no red pigment, as this will defeat the purpose. 
When a great deal of red is present, we have, of course, purple 
—another color which should not be used, as it looks black 
under most artificial light. 

Yellow and green, are little-used colors, but sometimes 
serve in curve-work as secondaries to black and red, as for 
example where it is desired to insert “quotas” or comparisons 
in fainter colors, on the same chart with black and red curves. 
These colors are also useful in map work, when areas are 
shaded, as intermediates between the red and blue extremes— 
the best sequence being red, orange, yellow, yellow-green, 
and blue-green. The other colors, brown, pink, and the like, 
are very little used, as they are not so distinctive as the simple 
primary colors. 


APPENDIX E 


OPTICAL ILLUSIONS IN CHARTS 


i BR eee os tl ae 
1} 
- 


i 
tn = : 


Wy 
iM 


. 
| 
hcl a ees a Le 


ie CHARTS AND GRAPHS 


is a powerful shading, as each case depends upon the surround- 
ing colors or shades with which it is in contrast. The chart- 
maker must in each case judge carefully of the effects of his 
shadings, and even if he cannot give equal emphasis to all 
parts of his chart, at least strive to avoid emphasizing the 
unimportant and slighting the important parts of its message. 

The accompanying illustrations show that, in the parti- 
cular case presented, white is more powerful than black and 
the white square appears larger than the black although the 
black has a border or outline added to it. They also show 
that a line may be made to look longer or shorter by the 
direction of arrows attached to it, that hatchings make the 
same rectangle look wider or narrower according to their 
direction, and also that bars hatched diagonally may be made 
to appear crooked instead of straight. These are but a few 
of the minor optical illusions against which the maker of the 
chart should be on guard. 


APPENDIX F 
THE VERBAL CHART 


It sometimes happens that business or economic data 
exceeds the powers of any popular chart; that while it can still 
be studied in the more advanced and technical charts, it can- 
not be presented to the layman by either bar-chart or amount- 
of-change curve. In such cases it is still often possible to 
fall back upon verbal pictures or descriptions which will 
conjure up in the reader’s mind some idea of the nature of 
this data. Having, as it were, no chart-paper or black-board 
large enough to picture the situation, we fall back upon the 
screen of the mind’s-eye, and seek, even though without 
detail or accuracy, to project the picture thereon. 

This has sometimes been done by astronomers in describing 
the dimensions of the solar system or popularizing other 
astronomical data. The important thing, of course, is to 
find familiar similes and, if possible, such as will make the 
salient features of the data even more noticeable. In the 
following example, this has been done by using the analogy 
of plants and trees, as these have only a limited height and 
emphasize the exceptional heights described: 

“The range of modern incomes baffles the imagination. 
It is impossible at the same moment to visualize the very 
small incomes of the great mass of families and the large 
fortunes enjoyed by a small handful of them. The variation 
is too great, both as to size of incomes and number of families. 

“Imagine a round field, one mile across, in every square 
foot of which one plant is growing. ‘The plants, about twenty- 
two million in number, will represent approximately the 
families of the country, each plant for one family; and we can 
assume that the height of the plant indicates the annual 
income of the family, one foot for each thousand dollars, and 
that the tallest plants are clustered together in the center of 
the field. Let us start at the edge of this circular field and 


7%3 


714 CHARTS AND GRAPHS 


walk the distance of a half-mile, to the center, observing the 
height of the plants which we pass on the way. 

“For more than three-quarters of our walk we can only 
speculate on the height of the plants we pass, for lack of any 
definite information from any source. If the tax-collector 
catches even the majority of those who have income of three 
thousand dollars or more, none of the plants thus far will 
reach up a yard from the ground. On the basis of the incomes 
he does report, however, we can make intelligent speculations, 
one of which is that the first half of our walk will be among 
plants puny and ill-nourished, reaching about our ankles. 
At 300 yards from the center we reach what is called the 
“upper tenth,” the plants growing nearly to our knees. At 
200 yards from the center they stand up two feet and at 150 
yards they will be a yard high. 

“We now enter the inner cluster of plants, comprising two 
or three per cent of the total, and with definite information 
from the government we Waich a rapid increase in height. 
Thus at 300 feet from the center the plants climb to five feet, 
at 200 feet they rise overhead ten feet from the ground, at 
100 feet they reach up thirty feet, at 50 feet they tower a 
hundred feet high. At thirty feet from the center they climb 
to two hundred feet, at 15 to five hundred, at 9 to a thousand 
feet high. These last represent nearly 300 families whose 
income exceeds a million dollars a year. In the center of 
these, extending far out of sight about a dozen rise to heights 
of which we know little except that they stretch at least a 
mile into the sky.”! 


1The example given is from remarks delivered before the American Statis- 
tical Association, December, 1920, and is not, of course, so complete a description as 
could be given in view of later tax-reports and investigations. 


4 
4 
‘4 


SHORT BIBLIOGRAPHY 


Bowley, A. L.: “Elements of Statistics.” 

Brinton, Willard C.: “Graphic Presentation of Facts.” 
Clark, Wallace: ““The Gantt Chart.” 

Fisher, Irving: ‘“The Ratio Chart.” 

Haskell, Allen C.: ‘How to Make and Use Graphic Charts.” 
Kelley, Trueman: “Statistical Method.” 

Knog, Wilford I.: “Elements of Statistical Method.” 
Peddle, John B.: “Construction of Graphical Charts.” 
Secrist, Horace: ‘‘Introduction to Statistical Methods.” 
Running, Theodore R.: ‘Empirical Formulas.” 

Yule, G. Udney: “Theory of Statistics.” 


As) 


e 


anus nia : 


ce)? 

4! 

ee rus (a 
21 L Wes 


INDEX OF PERSONS AND SOURCES 


A 


Andrews, E. A., 530 

Annalist, N. Y., 201 

Audit Bureau of Circulations, 56 
Ayer and Son, N. W., 84 

Ayers, Leonard, viii, ix, 207, 209, 211 


B 


Barth, Carl G., 580, 581 
Bartholomew’s Atlas of the World, 3 
Boston Board of Health, 241 
Bowley, A. L., ix, 49, 314, 434, 477 
Briggs, Henry, 371 

Brinton, Willard C., ix, 87, 252, 635 
Brissendon, Paul F., ix, 351, 467 
Burnet, Arthur R., ix, 252-260 


C 


Chaddock, Robert E., 348 

Chase, Stuart, ix 

Children’s Bureau, 335 

Clark, Wallace, viii, 35, 252, 270-277 
Cleveland Trust Company Bulletin, vui 
Codex Book Company, 458 

Country Gentleman, 645 


D 


Day, Edmund E., 53, 298 
Davenport, C. B., 331 

Dodge Co., F. W., 235, 237 
Doggett, Leonard A., 205 

Douglas, Paul and Dorothy, ix, 127 
Dublin, Louis I., 332, 343 


E 


Electric Railway Journal, 669 
“Elimination of Waste,” 142 


F 


Federal Reserve Bank of New York, vin, 
94, 95, 108, 116, 120, 141, 144, 
159, 163, 200, 202, 210, 211, 248, 
284, 291, 296, 297, 299, 302, 304, 
305, 306, 403, 615 

Federal Reserve Bulletin, 331 

Federal Trade Commission, 18, 19 


717 


Fisher, Irving, 298, 387, 413-415, 620 621 
Florence, P. Sargent, 312, 340 
Friedman, E. M., 112 


G 
Gantt, Henry L., vii, 34, 106, 261, 277, 
622 
Gerstenberg, Charles W., 43 
Gilbreth, Frank B. and L. M., 32, 36 


Grolier Society, 711, 712 
Gulick, Sidney, 16 


H 


Hale, Robert Lee, ix 

Hammond Map Co., 5 

Hansen, A. H., 214 

Harvard Bureau of Business Research, 
95, 159 

Haskell, Allen C., ix, 530, 531, 589, 639, 
669 

Hood, B. B., 531 

Hopper, A. R., 556 


I 


Industrial Commission of Ohio, 353, 354, 
606 
International Labor Office, 103, 196, 423 


Interstate Commerce Commission, 160 


K 


Kelley, T. L., ix, 49 

Keuffel and Esser, 186, 579, 581, 692 
King, Wilford I., ix, 49, 314, 326, 337 
Korzybski, Alfred, 377 


IE, 


La Fontaine, Henri, 13 
Lipka, Joseph, 1x, 493, 505, 528, 534, 
536, 554, 555, 562 


M 


MacCauley, Frederick R., 1x, 468 
Marshall, Alfred, 384 

Marx, Guido, 639 

Mercator, Gerardus, 4 
Merriman, 480 


718 INDEX OF PERSONS AND SOURCES 


Miller, Fred J., viii 
Mitchell, Wesley C., 240, 298, 463 
Moore, H. L., 240 


N 

Napier, John, 371 

National Automobile Chamber of Com- 
merce, 129, 162, 198 

National Board of Fire Underwriters, 
105, 311, 349 

National Bureau of Economic Research, 
359, 361, 478 

New York City Dept. of Health, 418 

New York Journal of Commerce, 180, 181, 
220-234, 248, 254 

New York State Dept. of Labor, 200, 
303, 404-406 


O 


Oficial U. S. Bulletin, 138 
Ogburn, William Fielding, viii, 377 
Outwater, Olive E., 442 


FP 


Pareto, Wilfredo, 446, 478 

Peddle, John B., ix, 534, 634-636, 640- 
642, 658 

Pogue, Joseph E., ix, 197, 218, 300, 424 

Polakov, Walter N., viii, 28, 106, 235, 
27M; 271s Sly O85 

Prentice-Hall, Inc., 576 

Prescott, Raymond B., 485 

Printer’s Ink, 206, 208 

Publisher's Weekly, N. Y., 121 


R 


Rand, McNally & Co., 4, 40, 41 

Red Cross, 56 

Register General of England & Wales, 427 
Robinson, T. R., 252 

Rorty, Malcolm C., x, 30, 31, 238, 591 
Running, Theodore R., ix, 483, 534 
Russian Information and Review, Lon- 


don, 90, 150, 151 


Scott, R. E., 656 

Scoville, John C., ix, 260 

Secrist, Horace, ix, 49, 164, 230, 314 
Snyder, Carl, viii 

-Standard Statistics Co., 204 
Steinmetz, Charles P., 370, 701 
Stewart, Ethelbert, 428 

Swift and Co,, 97 


as 


Thompson’s Outline of Science, 20 
Thurston, Robert, 658 
Tuskegee Institute, 418 


U 


United States Bureau of Census, 54, 56, 
59, 61, 81, 99, 107, 402, 438, 624, 
646 
United States Bureau of Education, 
315-318, 334, 431, 460, 461 
United States Bureau of Fisheries, 309 
United States Bureau of Labor Statis- 
tics, 36, 86, 87, 109, 126, 132, 137, 
203, 212, 239, 242, 244, 290, 299, 
301, 303, 309, 320, 324, 327, 337, 
352,) 388,: 7390; 3915 1395, Soke 
420, 432, 436, 445, 464, 474, 
475, 592, 600-605, 611 
United States Bureau of Mines, 285, 287 
United States Collector of Internal 
Revenue, 447, 470, 616, 618 
United States Commissioner General of 
Immigration, 122 
United States Department of Agricul- 
ture, 404-406 
United States Department of Commerce, 
vill, 182, 183, 243, 245, 286, 289, 
292, 488 
nited States Federal Trade Commis- 
sion, 19, 20 
nited States Public Health Service, 
158, 310 ; 
nited States Senatorial Investigating 
Committee, 101 


So 


Ss 


Vv 
Van de Wall, Constant, 2, 128 


Ww 


Webster, Richard, ix, 29 

Wenzel, John, ix, 136, 137, 154, 155, 184, 
260, 390, 393, 422 

World Almanac, 97 


Ne 


Yearbook of Churches, 106 

Yule, G. Udney, ix, 49, 314, 319, 331, 338, 
345, 350, 427, 430, 437, 441, 453, 
653, 660 


INDEX OF ILLUSTRATIONS BY SUBJECT-MATTER 


A 


Accident mortality rates, by age and sex, 
139, 409 
female, by age and cause, 340, 409 
in coal-mining, 109 
in specified industries, 137 
in warfare, 85 
male, by age and cause, 140, 409 
Accidents, from automobiles, 418 
daily cycle, by hours, 242 
in industry, classified by nature and 
place, 143 
in streets, 418 
Accident table, an American, 443 
Advertising, in magazines, 206, 208 
American Expeditionary Force, casual- 
ties, 85 
flow of supplies, 30 
number of soldiers, 207 
Animal life, evolution of, 20 
Audit Bureau of Circulations, state- 
groups of, 56 
Automobile, accidents, 418 
production, passenger cars and trucks, 
162, 198 
production, total, 129 


B 


Bank deposits, turnover rate, 403 
salaries, 333 
Banks, national, 128 
savings, U. S., 61 
savings, world by countries, 131 
Belting, leather, 580 
Bonds, prices, 159 
sales, 211 
yields, 576 
Book publishing costs, 583, 584 
Books published, in U.S. and England, 
121 
in world by countries, 115 
Budget, family, 1913, 86 
1913-1921, 87, 212 
philanthropic, 127 
Buildings, 236, 237 
Business cycle, the, 238 
Business, failures in, 99, 111 
Business organization, structure of, 17 
flow of goods and money in, 45 


C 


Cables, mileage in world, 483, 484 


Call money rates, 204, 284 
Capital in new incorporations, 180, 220- 
234, 248-254 
Carterian coordinates, 53-72 
Cattle, U. S., map of, 646 
Chain-store sales, 296, 297 
Chart-paper, samples of, 7, 184, 390, 458 
Charts, classification of, 14 
Chess openings, classification of, 27 
Chicago traffic map, 669 
Children, stature of, 335 
Cigar chain-store sales, 297 
Circle, calibration of, 96 
City finances, 308 
Civil War compared with World War; 
patents, 189 
prices, 163 
wages, 301 
Clothes, cost analysis of, 94 
Clothing store, cost analysis of, 95 
sales of, 297 
Cloudiness of skies, 430 
Coal, production, U. S., 128 
production, world, 417 
production, causes of failure, 106 
reserves unmined of the world, 113 
by countries, 608, 609 
Coal-mining, accidents in, 109 
College, students in, 128, 412 
teaching salaries in, 315-318, 334, 431, 
460, 461 
Commerce of the world, 483, 484 


> Commercial paper interest rates, 202 


Co-ordinates, 11, 53-72 
Copper, alloys, 658 
prices, 395 
Corporation, incomes, 618 
new, capital in, 180, 220-234, 248-254 
Cost 2nalysis, of retail stores, 95 
of specified articles of clothing, 94 
of Swift and Company, 97 
Cost of living, 1920, 132 
1913-1921, 92, 212 


720 


Cotton production, U. S., 128 
world, 417 
Curve charts, steps in making, 22-25 
Cycle, business, 238 
in department store sales, annual, 38 
in department store sales, weekly, 37 


D 


Death-rates, accident, 340, 409 
female, 439 
in coal mining, 109 
in specified industries, 137 
in the World War, 138 
ratio of male to female, 427 
violent, 418 
Deaths, in World War, 85 
Debt, public, U. S., 104 
public, world, 112 
Denmark, rents, 309 
Density of population, U. S., 26 
world by continents, 100 
Department-store, cost analysis, 95 
sales, cycles, annual, 38 | 
sales, cycles, weekly, 37 
sales, index, 296, 297 
Diphtheria antitoxin, 156, 310 
Divorces, 410 
Drug, chain-store sales, 297 
store cost analysis, 95 
Dry goods, chain-store sales, 297 


ig 


Earnings, bank employes, 333 
college teachers, 315-318, 334, 431, 
* 460, 461 
corporation, 618 
workers, 108 
see also ‘““‘Wages” 
Education, illiteracy, 59, 60 
students in colleges, etc., 128, 412 
truancy in schools, 667 
see also “‘College-teaching salaries” 
Eggs, prices, 239 
production, 245 
storage, cold, 243, 488 
Eight-hour turns in continuous-operation 
industries, 36 
Emigrants, 122 
Employment, in N. Y. factories, 200, 303 
in U. S., 200 
in world, 474 
length of, 351 
Evolution of animal life, 20 


INDEX OF ILLUSTRATIONS 


Exports, destinations of, 119, 217 
nature, 215 
ports of, 215 
see also “Foreign trade” 


F 


Factories, size of, 328 
Failures, in business, 99, 111 
in business and prices, 304 
Families, size of, 331, 345, 350 
Family budget, 1913, 86 
1913-1921, 87, 212 
Farm, land, 624 
property, 102 
wages, 404-406 
Farms, size of, 321, 323, 341-344, 435, 459 
Fatigue and output of workers, 158 
Financing, foreign, 141 
Fire, causes of, 105 
losses from, 181, 311, 313, 319, 328, 
336, 348, 349 
Five-and-ten-cent-store sales, 297 
Flow of goods, in a business concern, 279 
in an industry, 278 
Food, prices, 92, 126, 203,290, 303, 398, 420 
production of, 298 
retail establishments, 93 
values (caloric), 591 
Foreign exchange rates, 116 
and wholesale prices, 305 3 
Foreign trade, U. S., 83, 86, 115, 281 
world, 122, 127 
Furniture stores, cost analysis, 95 


- 


G 


Gantt charts, logic of, 28 
Garden-planting time in U. S., 46 
Gas-engine mixtures, 634 
Gasolene, consumption, 218 

stocks, 285, 287 
Gold, production of the world, 321, 322, 

400, 417 

reserves of the world, 116 

reserves ratio to paper currency, 135 
Grocery chain-store sales, 297 

store cost analysis, 95 


H 


Hardware-store cost analysis, 95 
Height, of children, 335 

of fathers and sons, correlated, 653 
Homicides, 418 
Hours of labor, 329, 346, 347 
Humidity, 650, 654, 657 


INDEX OF ILLUSTRATIONS 


I 


Illiteracy, by age, sex, and race, 59, 60 
Immigrants, by sex, 122 
Immigration, annually from 1860, 117 
by decades from 1800, and by country 
of source, 118 
Imports, by continent of source, 119, 216 
by port of entry, 86 
see also “‘Foreign Trade” 
Income and Expenditure, round flow of 
money, 31 
Incomes, distribution of, 361, 447, 470, 
616, 713 
Incorporations, capital in new, 180, 220- 
234, 248-254 
Industrial accident rates, 137, 143, 242 
Industry, waste in, 142 
Interest rates, and discount rates, 210 
and renewal rates, 211 
on call money, 204, 284 
on commercial paper, 202, 248 
Invention and warfare, 199 
Inventory analysis, 35 
Iron, production of pig in U. S., 1870- 
1920, 128 
production of pig in U. S., by states, 
1920, 57 
production of pig in world, 417 


Mi 


Jewelry-stores cost analysis, 95 
Jewish population of the world, 610 


L 


Labor, eight-hour shifts scheduled, 36 
turnover rates, 467 
wages, U.S. Steel Corp., 209 

Languages, number per continent, 91 
number of letters per word, 434 
population per language, 91 

League of Nations, 16 

Lettering pen, 692 

Letters per word, 434 

Library Volumes, 412 

Life expectancy, without tuberculosis, 

343 

influence of tuberculosis on, 332 

Living, costs, 86, 87, 92, 132, 212 
standards, 219 

Logarithms, table of, 374, 375 
geometrical construction of, 401 

Lynchings, 418 


721 


M 


Magazine, advertising, 206, 208 

date of issue, 84 

numbers published, 128, 412 
Mail-order-house sales, 297 
Manufacturing, index of volume of, 159, 

298 

Map, pictorial, 2 

population, 627 

of U.S., farm land, 624 

of U. S., garden-planting times, 46 

of U. S., natural resources, 2 

of world, elliptical, 5 

of world, heart-shaped, 3 

of world, hemispherical, 4 

of world, homolographic, 5 

of world, Mercator’s projection, 4 
Marriages, duration of, 429 

number of, 410 
Married persons, ages of, 338 
Meat-packing industry, joint interests of 

19, 20 

Men’s clothing stores, cost analysis, 95 
Mercator’s projection of the world, 4 
Merchant marine, 128 
Money, in circulation, 128, 398 

quantity theory of, 620, 621 

round flow of, 31, 45 

stocks of, 54 
Mortality rates, accident, 139, 340, 409 

female, 439 

in coal-mining, 109 

in specified industries, 137 

in warfare, 138 

male, 140 

ratio of male to female, 427 

violent, 418 


N 


Newspapers, 412 
Non-mathematical’charts, £4 
Normal curve, 452 

Normal ogive, 455, 456 
Normal surface, 660 


O 


Occupations, U. S., in peace, 114, 600-605, 
6ll 
U. S., in war-time, 607 
world, 114 
Dil, consumption, 197, 424 
stock prices, 300 


pel 


Optical illusions, 711, 712 
Output and fatigue of workers, 158 


P 


Packers, joint interests of, 18, 19 
Patents issued, 199, 412 
Pen, lettering, 692 
Periodicals, date of issue, 84 
number of, 128, 412 
Petroleum prices, 300 
Philanthropic budget, 127 
Phonograph records, 558 
Pig iron, production, 57, 128, 417 
Polar co-ordinate rulings, 72 
Political campaign expenditures, 101 
Population, density of, U.S. by states, 26 
density of, world by continents, 100 
U. S., 1790-1920, 145-149, 414, 415 
U. &., class alignments, 120, 214 
U. S., religious denominations, 106 
U. S., urban, 123 
world, from 1800, 417 
world, by continents, 91 
world, by languages, 91 
world, by races, 91 
world, by religions, 91 
Population-map of U. S., 627 
Potatoes, prices, 388-391 
Power-plant calculations, 585 
Presidential campaign expenditures, 101 
Prices, food, 92, 126, 203, 290, 303, 398, 
420 
retail, 132 
wholesale, 292, 299 
wholesale, and business failures, 304 
-wholesale, and foreign exchange, 305 
wholesale, changes in, 463 
wholesale, in specified countries, 305, 
306, 475 
wholesale, in U. S. and England, 302 
wholesale, in world and civil wars, 163 
Printers’ type-sizes, 556 
Production, by coal miners, 428 
by workers, 312, 340 
by workers, compared with fatigue, 158 
in specified classes of factories, 328, 
356-364, 462-469 
in specified states, 134 
in U. S. compared with world, 136 
index of volume of, 159, 298 
of basic commodities, 131, 291 
of specified commodities, 286 
Property, farm, 102 
total, 128 


INDEX OF ILLUSTRATIONS 


Publications, books, 115, 121, 583, 584 
periodicals, §4, 128, 206, 208, 412 
Public, debt, U. S., 104 
debt, world, 112 
wealth, 128 


Q 
Quantity theory of money, 620, 621 


R 


Races, population by, 91 
Railroad, accidents from, 418 
income of, 160 
mileage, U. S., 128 
mileage, world, 483, 484 
Rate-of-change scale, geometrical con- 
struction of, 401 
“Real wages,” 219 
Red Cross, state groups of, 56 
Religion, populations, in U. S., 106 
populations, in world, 91 
Rent increases, 432 
Retail store cost analyses, 95 
Retail food stores, 93 
Rifle grenades, loading of, 32 
Russia, imports into, 90, 150, 151 


Ss 


Sailing ships, 483, 484 

Sales, analysis by pin-maps, 41 
cost analysis of, 94, 95, 97 
efforts, channels of, 29 
routing of salesmen, 40 

Salmon production, 309 

Savings bank deposits, in U. S., 61 
in world, 130 

Scallop-shells, 331 

Scarlet fever, 241 

School truancy, 667 

Ships, in U. S., 128 
in world, 144, 483, 484 

Shirts, cost analysis, 94 

Shoes, cost analysis, 94 

Shoe stores, cost analysis, 95 

Spelling, letters per word, 434 

Star-light, 466 

Statistical department, floor-plan, 10, 688 
routing of reports, 42 

Stock inventory analysis, 35 

Stock market prices, 204 
and sales, 201, 284 

Stocks of specified commodities, 289 

Stores, retail, cost analysis, 95 
retail, food, 93 


INDEX OF ILLUSTRATIONS 


Street accidents, 418 

Strikes, and lock-outs, 244 
duration of, 320, 324, 445, 464 
settlements of, 592 
sizes of, 327, 436 

Suicides, 418 

Suits of clothing, cost analysis, 94 

Swift and Company sales dollar, 97 


oe 


Telegraphs, mileage, 483, 484 

Textiles, production, 298 

Trade balance, U. S., 281 

Trade Union membership, world, 103, 

196, 423 

Truancy in schools, 667 

Trucks, speeds of, 329 

Tuberculosis, effect on longevity, 332 
life-expectancy without, 343 

Type sizes, 556 


U 


Unemployment, 474 
see also “Employment” 
Unions, trade, 103, 196, 423 
United Cigar Stores Co. sales, 182, 183 
United States, city finances, 308 
cultural growth, 412 
debt, 104 
density of population, 26 
foreign trade, 83, 86, 115, 119, 215-217, 
281 
illiteracy, 59, 60, 667 
map, of cattle, 646 
map, of farm-land, 624 
map of garden-planting times, 46 
map of land areas, 626 
map of natural resources, 2 
map of population, 627 
population, 145-149, 414, 415 
population, density, 26 
population, foreign-born, 107 
population, religious, 106 
population, urban, 123 
progress of, 128 
property wealth, 128 
publications, see ‘‘Publications” 
savings banks, 61 
wealth, 128 
World War cost, 97 


W 


Wages, common labor, 209 
farm and factory, 404-406 


723 


index of, 301, 303, 398 

in Ohio, 353, 354, 606 

of women, 354 
War, and invention, 199 

death rates in, 138 

see also “World War” 
Waste in industry, 142 
Wealth, public, 128 
Wholesale business sales, 615 
Winds, 578 
Women-workers, hours of labor, 352 

in France in war-time, 213 
Words, length of, 434 
Workers output, 312, 340 

and fatigue, 158 

in coal-mine, 428 
Workmen’s compensation, 337 


World, areas by continents, 91 
cables, 483, 484 
coal production, 417 
commerce, 483, 484 
cotton production, 417 
gold production, 321, 322, 400, 417 
gold reserves, 116, 135 
iron production, 417 
land and sea areas, 91 
languages, 91 
map, elliptical, 5 
map, heart-shaped, 3 
map, hemispherical, 4 
map, homolographic, 5 
map, Mercator’s projection, 4 
population, 417 
population, by continents, 91 
population, by languages, 91 
population, by races, 91 
population, by religions, 91 
population, density, 100 
production, 417 
railways, 483, 484 
shipping, 483, 484 
stocks of money, 54 
telegraphs, 483, 484 

World War, American casualties, 85 
A. E. F., flow of supplies, 30 
A. E. F., number of soldiers, 207 
cost to U. S., 97 
cost to all countries, 112 
occupations in time of, 607 
patents compared with Civil War, |199 
prices compared with Civil War, 163 
wages compared with Civil War, 301 
women-workers, French, 213 


GENERAL INDEX 


A 
Abscissae, 9, 69, 147, 151, 190 


axis of, 9 


position of plotted-points upon, 69 


(note) 
Absolute, band-chart, 212, 213 
stair-cased, 207 
Absolute data, 305, 681 
Abstract series, 150, 675 


Accounts payable, sales & payments, 


charts for, 278 
Adding-machines, 58 
Advertising, charts for, 89, 124, 689 
Alignment chart, 533 
Amount-of-change analysis, 385 
Amount-of-change curve, 386 
Amount-of-change scales, 679 
Analysis, amount-of-change, 385 
Analysis, light, 258, 295, 352, 481, 703 
Analysis, rate-of-change, 382 
Animated chart, 685 
Anti-logarithms, 373 
Anti-logarithmic projection, 476 
Architects’ rule, 183 
Area bar-charts, 613 
Area bars, 676 
Area chart, compound, 64 
Areal co-ordinates, 588 
Areas, 78, 598 
Areas, in pie-chart, 91 
Arithmetic change, 386 
Arithmetical progression, 377 
Arithmetical series, 377 
Arrays, in work-sheets, 58 
Asymptote, 428, 453, 505 
Average, 300 

compounded, 250 

moving, 233, 245 
Averages, 673 

method of, 480 
Automatic chart, mechanical, 237 
Axis, 9, 64, 533, 589 

dummy, 575 

scales for, 167 
“Axis of Abscissas,” 9 
“Axis of ordinates,’ 9 
Axonometric chart, 634, 638 


Axonometric drawing, 665 
Axonometric paper, 656 


B 


Balance-chart, 620 
Balance, income and outgo charts, 278 
Band chart, 212 
absolute, 212 
absolute stair-cased, 207 
frequency, 339 
percentage, see Band-chart, relative 
relative, 213 
relative, staircased, 215 
smoothed, relative, 214 
Bar, 100%, 83, 588 
classification chart and, 85 
position of data on, 87 
scale, 85 
Bar-chart and classification-chart, 107 
Bar-chart, field of, 101 
scale of, 101 
table and, 100 
Bar-charts, 99, 676, 691 
area, 613 
circular, 124 
composite, 111 
compound, 111 
compound absolute, 114 
compound relative, 114 
connecting lines, 118 
correlation, 116 
mechanical, 691 
mirroring, 116 
multiple, 115 
pictorial, 124 
pictorial horizontal, 117 
silhouette, 285 
subtotals in, 103 
typewriter, 108 
vertical, 134 
vertical, compound, 207 
Bars, 76, 99 
area, 676 
horizontal, 99 
silhouette, 473 
vertical, 331 
Base figure, 296 
Base-line omitted, 136, 155 


725 


726 GENERAL INDEX 


Base-line, wavy, 159 
Bead map, 665 
“Benday” process, 647, 663, 706 
Black-line process, 705 
Blue-prints, 43, 705 
Board, drawing, 691 
“‘Boiler-chart,” 29 

Box, light, 259 
Box-chart, 14, 16, 17 
Brown-line process, 705 
Bulletin-board, 22 
Business cycles, 238, 240 


C 


Calculating chart, 511, 533 
Calculators, mechanical, 687 
Calculators, pulley-wheel, 585 
Calibration of Circle, 96 
Captions, in work-sheets, 58, 60, 676 
Cartesian cor-ordinates, 71, 151 
Cartography, 6 (note) 
Chain-percentages, 307, 385, 419 
Chain-relatives, 385, 419 
Chance variation, 479 
Change, arithmetic, 386 

difference, 386 

geometric, 386 

increment, 386 

logarithmic, 386 

organic, 386 

percentage, 386 
Characteristic, 373 
Chart, alignment, 533 

animated, 685 

area, compound, 612 

axonometric, 634, 638 

balance, 620 

band, 212 

box-chart, 14, 16 

calculating, 511, 533 

classification, 13, 84 

composite non-mathematical, 39 

counter-poise, 620 

data separated from, 137, 163 

Gantt, 34 

Gantt idleness, 106 

Gantt progress, 622 

gun-shot, 204, 205, 334 

100%, circular, 89 

isometric, 639 

logarithmic retrospect, 474 

“marble-cake,” 605 

mechanical automatic, 237 

one-dimensional, 75, 83 


orthographic, 634, 642 
pipe-organ, 135 
planetary, 17, 18 
procedure, 25 
progress, vii, 261 
rate-of-change, 414, 421 
ratio, 414 
route, 21 
semi-logarithmic, 416 
formula for, 507 
sky-line, 617 
summary, 278, 280, 281, 677 
three-dimensional, 75 
time, 32, 33 
tree, 18 
tri-linear, 588 
two-dimensional, 75, 76, 152 
verbal, 713 
volumetric, 631 
Zee, 252 
Chart-fields, 699, 702 
standard, 183 
Chart-form, standard, 183 
Chart-modulus, 517, 537 
Chart-paper, 174, 175, 176, 177, 178, 
"179, 184 
part-deck, 394 
rate-of-change, 392, 394, 395 
split-deck, 394 
Chart-production, “Straight-line” meth- 
ods of, 696 
Charts, colors in, 709 
deceptive, 59, 77, 125, 133, 136, 155 
evolution of, vii, 144 
evolution of curves in, 147 
evolution of pie-chart, 89 
fanned out, 179 
for advertising, 89, 124, 689 
for commitments, orders, and receipts, 
278 
for consumption, production, and 
stocks, 278 
for cost-distribution, 97 
for income, outgo, and balance, 278 
for orders, receipts, and commitments, 
278 
for production, consumption, and stock, 
278 
for production, shipments, and stock, 
278 
for receipts, uses, and raw materials, 
278 
for sales, payments, and accounts pay» 
able, 278 
forward-looking, 261 


GENERAL INDEX 727 


implements for making, 691 
. lantern slides, 174 
large, 173 
letter-size, 173 
motion-pictures of, 685, 704 
photographs of, 174 
popular, 209, 216, 217, 356, 421, 611, 
622, 689 
sensational, 89, 127 
Chart-size, 182 
Chart-title, rules for, 699 
Circle, calibration of, 96 
hundred-per-cent, 89, 90, 596, 610, 676, 
691 
Circular bar-charts, 124 
Circular time-chart, 37 
Circular slide-rule, 581 
Classification charts, 13, 20, 84 
routing on, 45 
Classification-chart and bar-chart 107 
Classification-chart and complicated data, 
19 7 
Classification-chart and map, 46 
Classification-chart and 100% bar, 85 
Classification, decimal, 13 (note) 
methods, 13 
Classified column-headings, 59 
Classified stubs, 60 
Clock-chart, 37, 38, 236, 237 
Cloth, tracing, 703 
Collapsible model, 635, 655 
Colored inks, 691 
Colors, in charts, 709 
Columns, 53, 76 
Column-headings, 77, 676 
Column headings, classified, 59 
in work-sheets, 55 
Column symbols, 60, 61 
in work-sheets, 59 
Compass, 691 
Composite bar-charts, 111 
Composite charts, 39 
Composite curves, 198 
Composite distribution, 675 
Composite frequency curves, 338 
Composite nomographs, 560 
Compound area chart, 611 
Compound bar-charts, 111, 114, 207 
Compound frequency curves, 426 
Compounded Average, 250 
Computing, 698 
instructions, 61 
machinery, 693 
methods, straight-line, 53 
Connecting lines, 16 


bar-charts, 118 
Continuous data, 326 
Continuous frequency series, 326 
Continuous series, 677 
Contour lines, 642, 644, 657, 664 
Co-ordinate axes, 67 (note) 
Co-ordinate writings, 11, 63, 64, 66, 67 
Co-ordinates, 9, 63, 135, 152, 190 
areal, 588 
cartesian, 71, 151 
equal, 168 
nomenclature of, 11 
polar, 72 
step in making, 63, 64, 66, 67 
three-dimensional, 135 
triangular, 364 
Correlation, 294, 411, 481, 659 
bar-charts, 116 
Cost distribution, charts for, 97 
Counter-poise chart, 620 
Cross-hatching, 691, 711 
Cross-ruled paper, 7 
Crayons, 692 
Cubes, 80 
Cubic volume, 133 
Cumulation, 225, 255, 680 
Cumulation, frequency, 341, 444 
Cumulation, “‘less-than,”’ 342, 444 
Cumulation, “more-than,” 342, 444 
Cumulative, 225, 255, 341, 342, 444, 680 
Curve, amount-of-change, 386 
equation to the, 490 
exponential, 507 
frequency, 308, 326 
frequency, asymmetrical, 428 
frequency, composite, 338 
frequency, compound, 426 
frequency, J-shaped, 428 
frequency, ‘logarithmic, 426 
frequency, relative, 340 
frequency, symmetrical, 427 
frequency, U-shaped, 429 
frequency, Zoned, 338 
historical, 220 
Lorenz, 356 
of error, normal, 450, 479, 659 
periodic, 487 
probable, 451, 454, 680 
rate-of-change, 388, 389, 391 
rate-of-change historical, 416 
rate-of-change, range of variation, 402 
rectilinear, 205 
rounded, 336 
sine, 487 
smoothed, 207, 208, 230, 329, 677 


208 GENERAL INDEX 


staircase, 205, 206, 208, 329, 677 

staircased, pseudo, 209 

undulate, 487 
Curve-charts, pictorial, 217, 219 

position of data on, 163 

steps in making of, 33 

typewriting of, 166 
Curve-equating, 488 
Curve-field, 151 
Curve-fitting, 205, 477, 510 
Curves, 76, 145, 677 

composite, 199 

evolution of, 147 

for formulae, 511 

formulae for, 490 

French, 691 

scale-figures, 161, 164 
Cycle, 37, 235 

business, 238, 240 
Cyclic fluctuations, 240 
Cyclograph, 235, 237 


D 


Data, absolute, 305, 681 
continuous, 326 
discrete, 326 
flow, 279, 328 
frequency, 308 
frequency, moving total of, 318 
fund, 279, 328 
period 327 677 
periodic, 236 
point, 327, 333, 677 
point-and-period 329 
position on curve-chart 163 
posit: of on 100% bars, 87 
position of, on pie-chart, 93 
recurrent, 236 ; 
relative, €81 
reliability of, 477 
route-chait, tabulation of, 22 
separated from chart, 137, 163 
singular use of, 8 
stock, 328 
stream, 279, 328 
with the chart, 225 

Deceptive charts, 59, 77, 125, 133, 136, 

155 
Decils, 344, 427 
Dependent variable, 62, 153, 164, 314, 
333, 672 


Difficulties, labelling, 137, 164, 619 
typographical, 84, 137, 164, 619 
Dimensions, 64, 74, 152, 598 
in pie-chart, 91 
third, 630 
Discrete data, 326 
Discrete frequency series, 326 
Discrete series, 677 
Dispersion, 361, 427 
Distortion maps, 3, 7 
Distributions, 313, 673 
composite, 675 
Dividers, 691 
Double frequency series, 650, 652 
Double ogive, 657 
Double ruling pen, 691 
Drafting, 699 
Drawing, axonometric, 665 
board, 691 
Dummy axes, 575 


E 


Engineers’ rule, 183, 186 
Equal co-ordinates, 168 
Equa: scates, tield with, 68 
Equation of ordinary hyperbola, 501 
Equation of ordinary parabola, 505 
Equation to the curve, 490 
Equation: y = x, 490 
Error, curve of, 50, 479, 659 
Evolution of charts, vii, 144 

curves, 147 

pie-chart, 89 
Extrapolation, 196, 197, 413, 424, 510 


F 


“Fanned out” charts, 179 
Field, 47, 69, 154, 702, 709 
bar-chart, 101 
co-ordinates not perpendicular, 71 
curve, 151 
origin in corner, 70 
origin near one edge, 70 
origin not shown, 71 
position of, 223 
standard, 183 
standard, vertical scales, 187 


standard, vertical scale, Zee-chart, 258 


Field-investigations, 697 
Figure, base, 246 


Dependent variable, ogives, choice of, 348 Figures, flow, 677 


Diagrams, 1, 43 
Diagrams, pie, 676 


fund, 677 
index, 305 


4 
7 
; 


GENERAL INDEX 


per capita, 682 
relative, 305, 681 
stock, 677 
stream, 677 
Fitting curves, 477, 510 
Fixed rules, 577 
Floor-plans, 6, 43 
Flow data, 279, 328, 677 
Fluctuations, secular, 240 
cyclic, 240 
Forecast, 197, 294, 423 
Forest model, 635, 654 
Formula for semi-logarithmic chart, 507 
Formulae, 60 
curves for, 511 
for curves, 490 
in work-sheets, 59 
Forward-looking chart, 261 
Freak peak, 181 
French curves, 337, 691 
Frequency band-chart, 339 
Frequency curve, 308, 326 
composite, 338 
compound, 426 
extremely asymmetrical, 428 
J. shaped, 428 
logarithmic, 426 
moderately asymmetrical, 428 
relative, 340 
symmetrical, 427 
U-shaped, 429 
zoned, 338 
Frequency data, 308 
Frequency polygon, 329, 677 
Frequency series, 149, 308, 675 
continuous, 326 
cumulation of, 341, 444 
discrete, 326 
double, 650, 652 
historical, 652 
moving-total of, 318 
Frequency surfaces, 650 
“Function,” 61 
Fund data, 227, 279, 328, 433, 677 


G 


Gantt chart, 34, 35, 260 

Gantt idleness chart, 106 

Gantt progress chart, 260, 622, 680 
Geographical series, 150, 675 
Geometric intervals, 437 
Geometrical progression, 377 
Geometrical series, 377 

Gilbreth process-chart, 32 


729 


Gompertz curve, 381, 483 

Graduate variates, 326 

Growth, organic, law of, 371, 388, 507 
“Gun-shot” chart, 204, 205, 334 


H 


“Half-tone,” 705 

Hectograph, 706 

Histogram, 205, 220, 329 

Historical curves, 220 

Historical frequency series, 652 

Historical rate-of-charge curves, 416 

Historical series, 149, 220, 675 

Historigram, 220 

Horizontal bar-charts, pictorial, 117 

Horizontal scale, 172 

Horizontal scales, facsimile typewriting, 
173 

Hundred-per-cent bar, 83, 588, 676 

Hundred-per-cent circle, 89, 596, 610, 676 

Hundred-per-cent rectangle, 604 

Hundred-per-cent squares, 598 

Hundred-per-cent triangles, 588, 658 

Hyperbola, ordinary, equation of, 501 


Idleness-chart, 106 
Illusions, optical, 617, 711 
Implements for making charts, 691 
Income, outgo, and balance; charts for, 
278 
Incomes, distribution of, 446, 478, 713 
Pareto’s law of distribution of, 446, 478 
Increment charge, 386 
Independent variable, 62, 153, 314, 672 
Index numbers, 294, 305, 403, 681 
India ink, 691 
Ink, colored, 691 
India, 691 
Integral variates, 326 
Interpolation, 195, 196, 510 
Interest point, 152 
Intervals, geometric, 437 
Investigations, field, 697 
Isometric chart, 44, 639, 656 
Isopleth, 533 
Items, position of, on work-sheets, 58 


K 


Keys, 6, 11, 668 
Knives, 692 


13° 
L 


Labelling, difficulties in, 93, 137, 164, 619, 

699, 708 
Lantern-slides for charts, 174, 703 
Large charts, 173 
Law of incomes, Pareto’s, 478 
Law of organic growth, 371, 386, 416, 507 
Least squares, method of, 480 
“Less-than” cumulation, 342, 444 
Lettering pen, Payzant, 692 
Letter-size charts, 173 
Light analysis, 238, 295, 350, 413, 481, 703 
Light box, 259, 707 
Line-cuts, 705 
Lines, contour, 642, 644, 657, 664 

orthographic, 657 
“Link-relative,” 307, 385, 419 
Logarithmic frequency curves, 426 
Logarithmic ogives, 444 
Logarithmic progression, 384 
Logarithmic projection, reasons for, fre- 

quency series, 433 

reasons for, historical series, 415 

retrospective, 473 
Logarithmic retrospect chart, 474 
Logarithms, 370, 373 

table of, 374, 375 
Loglogs, 371, 443, 476 
Lorenz curves, 356 


M 


Machinery, computing, 693 
Mantissa, 373 
Map, 1, 39, 658, 676, 692, 702 
and classification-chart, 46 
bead, 665 
celluloid-coated, 42 
dot, 663 
of United States, pictorial, 2 
of world, ‘Butterfly map,” 5 
of world, elliptical, 5 
of world, heart-shaped, 3 
of world, hemispherical, 5 
of world, homolographic, projection, 5 
of world, Mercator’s projection, 4 
of world, “‘orange-peel projection,” 5 
pin-maps, 40, 663, 667 
population, 623 
relief, 661 
route-map, 40 
stereographic, 665 
string, 42, 668 
topographical, 664 


GENERAL INDEX 


tree, 666 
Zone, 664 
Map-keys, 11 
Map-mounting, 41 
Map-projections, 5 
Maps, distortion in, 3, 5 
of United States, 1 
of small areas, 6 
of the world, 3, 5 
variety of, 1 
Map-tacks, 40, 663 
“‘Marble-cake chart,”’ 605 
Measurement, units of, 674 
Mechanical automatic chart, 237 (note) 
Mechanical bar-charts, 691 
Mechanical calculators, 687 
Median, 343 
Mercator’s projection, 4 
Method of averages, 480 
Method of least squares, 480 
Method of selected points, 480 
Methods, computing, straight-line, 53 
statistical, 49 
Mimeograph, 706 
Mimeoscope, 707 
Mirroring, 11, 70, 116 
Mode, 343, 680, 
Model, 630, 631, 635, 654, 663 
collapsible, 635, 655 
forest, 635, 654 
solid, 654 
Moduli, scale, 517, 536 
Modulus, 516, 517, 537 
“More-than” cumulation, 342, 444 
Motion-picture chart, 685, 704 
Moving average, 233, 244 
Moving total, 228, 245, 255, 680 
of frequency data, 318 
plotting points, 231 
taper-smoothed, 230 
Multiple bar-chart, 115 
Multigraph, 706 


N 


Nomogram, 533 
Nomograph, 533 
parallel, 533 
composite, 560 
zigzag, 560 
Normal, 246 
Normal curve of error, 450, 479, 659 
Normal ogive, 454 
Normal probabilities, curve of, 451 
Numbers, index, 294, 305, 403, 681 


GENERAL INDEX 


Oo 


Ogive, 341 
choice of dependent variable, 348 
double, 657 
logarithmic, 444 
normal, 454 
Omitted zero-line, 136, 155 
One-dimensional chart, 75, 83 
Orders, receipts and commitments, chart 
for, 278 
Ordinates, 9, 69, 147, 151, 190 
axis of, 9 
tri-axial, 364 
Organic growth, law of, 377, 388, 416, 507 
Origin, point of, 8, 69 
Orthographic chart, 634, 642 
Orthographic lines, 657 
Orthographic rulings, 664 
Optical illusions, 77, 87, 93, 100, 617, 711 
Outgo, income, and balance, charts for, 


F 


Pantograph, 692 
Paper, axonometric, 656 
Parabola, ordinary, 505 
Parallel nomographs, 533 
Pareto’s law of incomes, 446, 478 
Part-deck chart paper, 394 
Payments, sales and accounts payable, 
charts for, 278 

Payzant lettering pen, 692 
Peak, freak, 181 
Pen, double-ruling, 691 

lettering, Payzant, 692 

railroad, 691 

ruling, 691 

ruling, curved, 692 
Per capita figure, 682 
Percentage, chain, 307 
Percentage change, 386 
Percentiles, 344, 427 
Period data, 327, 677 
Periodic curves, 487 
Personnel, statistical, 700 
Perspective, 638 
Photo-engraving, 705 
Photographs of charts, 174, 705 
Photostats of charts, 173, 705 
Pictogram, 221, 308 
Pictorial bar-charts, 124 
Pictorial curve-charts, 217 
Pictorial horizontal bar-charts, 117 


731 


Pie-charts, 89, 596, 610 
area in, 91 
dimensions in, 91 
labelling, 93 
in metal, 97, Fig. 80 
position of data on, 93 
psychological appeal of, 97 
scale, 93 
Pie diagrams, 676 _ 
Pin-maps, 40, 41, 667 
Pipe-organ chart, 135, 207 
Planimeter, 330, 692 
Planetary-chart, 17, 18 
Plotting, 47, 699 
in spaces or upon ordinates, 191, 192, 
193 
moving totals, 231 
Plotting-points, 190 
on abscissae, 69 (note) 
Point data, 327, 333, 677 
Point-and-period data, 329, Figs. 287-288 
Point of origin, 8 
Point of reference, 6 
Points, selected, method of, 480 
Polar co-ordinates, 72 
Polygon, frequency, 329, 677 
Polynomial, 505 
Population, 313 
Population maps, 623 
Popular charts, 209, 216, 217, 356, 421, 
611, 622, 686, 689 
Popular rate-of-change charts, 421 
Powers projection, 475, 497 
Presentation, popular, 686 
Probable curve, 451, 454, 680 
Procedure-chart, 25 
Process, Benday, 647, 663, 706 
black-line, 705 
brown-line, 705 
Process-chart, 32, 36 
Process, Van Dyke, 705 
Processes, straight-line, in chart making, 
44, 692 
Product, 598 
Production, consumption and_ stocks, 
charts for, 228 
shipments and stock, charts for, 278 
Progress charts, vii, 261 
Progression, 377 
arithmetical, 377 
geometrical, 377 
logarithmic, 384 
Progressive average, see moving average 
Progressive totals, see moving total 
Projection, object of special scales, 492 


7132 


powers, 497 

probabilities, 454 

reciprocal, 497 

retrospective logarithmic, 473 
Projections, scale, special, 482 
Projection, square, 497 

square-root, 482, 497 
Protractor, 96, 582, 691 
Pseudo-staircased curve, 211 
Psychological appeal of pie-charts, 97 
Pulley-wheel calculators, 585 


Q 
Quadrants, 9 
“Quadrille” paper, 7 
Quartiles, 344, 427 
Questionnaire, 697 
Quotas, 249, 273, 710 
calculation of, 249 


R 


Railroad pen, 691 
Range, 427, 432 
Rate-of-change analysis, 382 
Rate-of-change chart paper, 392, 394, 395 
Rate-of-change charts, popular, 421 
zero in, 407 
Rate-of-change curves, historical, 416 
Rate-of-change curve, 388, 389, 391 
range of variation, 402 
Rate-of-change scales, 387, 397, 679 
Ratio-charts, 414 
Raw materials, receipts and uses, charts 
for, 278 
Receipts, orders & commitments, charts 
for, 278 
Receipts, uses and raw materials, charts 
for, 278 
Reciprocal projection, 497 
Record, time, 34 
Rectangle, hundred-per-cent, 604 
Rectilinear curve, 205 
Relative band-chart, 213 
smoothed, 214 
staircased, 215 
Relative figures, 305, 681 
Relative frequency curve, 340 
Reliability of data, 477 
Relief maps, 661 
Report, rules for, 701 
statistical, 83 
Research, 696 
Residuals, 480 
Retrospect chart, logarithmic, 474 


GENERAL INDEX — 


Retrospective logarithmic projection, 47 
Ribbons, 43 
Rounded curves, 336 
Route-charts, 21 
Route-map, 40 
Routing on classification-chart, 45 
Row, 76 
Rows, in work-sheets, 58 
Rule, architects’, 183 
Engineers’, 183, 186 
slide, circular, 581 
stationary, 577 
Rules, fixed, 577 
slide, 577 
Ruling, co-ordinate, 11, Fig. 11 
Rulings, orthographic, 664 
Ruling pen, 691 
curved, 692 


S 


Sales, payments and accounts payable. 
charts for, 278 
Sampling, 313, 477 
Scale, 8, 167, 515, 533 7 
Scale of amount-of-change chart, 679 
Scale of bar-chart, 101 . 
Scale of 100% bar, 85 | 
Scale of pie-chart, 93 
Scale of rate-of-change chart, 387, 397, 
679 
Scale of summary chart, 283 
Scale-figures, curves, 161, 164 
Scale-moduhi, 517, 536 
Scale projections, special, 482 
Scale-projection, special, object of, 492 
Scales, convenient vertical, 187, Fig. 169 
charts for selection of, 524, 557 
engineers’ ruler, 186 
horizontal, 172, 191 
horizontal, facsimile typewriting, 173 
slide-rule for, 397 
triangulation method of projection, 
183, 397, 553 
typewriting, 172 
vertical, 177, 186, 188, 258 
for Zee-charts, 258 
Scatteration, 311 
Schedules, 249 
Seasonal, 223 
Section-liner, 691 
Secular fluctuations, 240 
Secular trend, 246 
Selected points, method of, 480 
“Semi-logarithmic” charts, 416 
formula for, 507 


y 


Sensational charts, 89, 127 
Sepia print, 205 
Series, 9, 62, 77, 679 
arithmetical, 377 
abstract, 150, 675 
double frequency, 652 
discrete, 677 
continuous, 677 
frequency, 149, 308, 675 
frequency, continuous, 321 
frequency, cumulation of, 341 
frequency, discrete, 321 
frequency, double, 65 
geographical, 150, 675 
geometrical, 377 
historical, 149, 220, 675 
historical frequency, 652 
Shadings, 86, 617, 647, 657, 661, 709, 711 
Shifting zero point, 476 
Shipments, production and stock, charts 
for, 278 
Silhouette bar-charts, 285, 473 
Sine curve, 487 
Skewness, 365, 428 
Sky-line chart, 617 
Slide rule, 577, 695 
circular, 581 
Slide-rule for scales, 397 
Slides, lantern, 703 
Smoothed curve, 207, 214, 230, 329, 677 
Smoothed surface, 642, 653 
Solid, 631, 654 
Space, dimensions for charting, 75 
Special scale-projection, object of, 492 
“Split-deck” chart paper, 394 
Square, hundred-per-cent, 598 
Square-root projection, 497, 482 
Squares, method of least, 480 
Squares projection, 475, 497 
Staircase curve, 205, 206, 207, 215, 329, 
677 
Staircased curve, pseudo, 209 
Staircased surface, 642, 653 
Standard chart-form, 183, 185, 187 
States, arrangement of, 55 
Stationary rule, 577 
Statistics, 48, 671, 696 
Statistical department floor-plan, 10,688 
Statistical department personnel, 700 
Statistical methods, 49 
Statistical report, 83, 700 
Stereographic map, 668° 
Stereographs, 634 
Stock data, 279, 328, 433 
Stock figures, 677 


GENERAL INDEX 


433 


Stock, production and shipments, charts 
for, 278 
Stocks, 227 
production & consumption, charts for, 
278 
Straight-edge, 692 
Straight-line computing methods, 53 
Straight-line processes, 44 
Stream chart, 119, 620 
Stream data, 279, 328 
Stream figures, 677 
Streams, 227 
String-maps, 668 
Stubs, 58, 76, 153, 676 
Sub-total, 15 
Sub-totals in bar-charts, 103 
Summary chart, 278, 280, 677 
vertical scale, 283 
Surface area, 133 
Surface, frequency, 650 
smoothed, 642, 653 
staircased, 642, 653 
Symmetry, 477 
System, Gantt, 34 
ly 
Table, 53, 57, 58, 698 
Table and bar-chart, 100 
Tabulations, 53, 57, 58, 310, 698 
Tabulation, position of total in, 58 
Taper-smoothed moving total, 230 (note) 
Three-dimensional chart, 75, 76, 630 
Time-chart, 32, 36, 37 
Time record, 34 
Title, 699 
Topographical map, 664 
Total, moving, 245, 680 
position of, in a tabulation, 58 
Totals, 673 
Tracing cloth, 703 
Tree-chart, 18 
Tree-map, 666 
Trend, 230, 245 
secular, 246 
Triangles, hundred-per-cent, 588, 658 
Triangular co-ordinates, 364 
Triangulation sheets for scales, 553 
Tri-axial ordinates, 364 
Tri-linear chart, 588 
Tri-squares, 691 
T-square, 691 
Two-dimensional chart, 75 
Typewriters, 693 
Typewritten bar-charts, 108 


States, maps of, 1 
f measurement, 674 
TSC, oe 


eee 62, "153, 164, 314, 333, 672 

is independent, 62, 153, 314, 672 
‘Variates, graduated, 326 

integral, 326 

_ “Variations, chance,” 4795 

Vat-chart, 30 ; 

Vertical scale, 177, 188 

Vertical bar-charts, 134, 331 

Volume statistics, 674 


Volumetric chart, 631 


W 


Weighting, 249, 250 

Work-sheets, 53, 698 
adding machines, 58 
arrays, 58 


SX, 9, 62, OF 


“ras, 65, 66, 135, ae 153, 154, 


aS 


CY eon Gae OO: ‘ 
“Y”-axis, 66, 135, 151, 153, 651 


Z 


(eae 9, 62 
“Z’-axis, 71, 651 
Z-ordinates, 651 
Zee-charts, 252, 680 
Zee-chart, scales for, 258 
Zero, 432 
arbitrary, 156 
Zero in rate-of-change charts, 407 © 
Zero-point, shifting of the, 493 
Zero-line, heavy, 160 
omitted, 136, 155 
Zigzag nomographs, 560 
Zone-curve, 202, 204 
Zoned frequency curve, 338 
Zone-map, 664 


peeeeestons 


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